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Dual graph

In the mathematical discipline of graph theory, the dual graph of a planar graph $G$ is a graph that has a vertex for each face of $G$. The dual graph has an edge for each pair of faces in $G$ that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge $e$ of $G$ has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of $e$. The definition of the dual depends on the choice of embedding of the graph $G$, so it is a property of plane graphs (graphs that are already embedded in the plane) rather than planar graphs (graphs that may be embedded but for which the embedding is not yet known). For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph.

Historically, the first form of graph duality to be recognized was the association of the Platonic solids into pairs of dual polyhedra. Graph duality is a topological generalization of the geometric concepts of dual polyhedra and dual tessellations, and is in turn generalized combinatorially by the concept of a dual matroid. Variations of planar graph duality include a version of duality for directed graphs, and duality for graphs embedded onto non-planar two-dimensional surfaces.

These notions of dual graphs should not be confused with a different notion, the edge-to-vertex dual or line graph of a graph.

The term dual is used because the property of being a dual graph is symmetric, meaning that if $H$ is a dual of a connected graph $G$, then $G$ is a dual of $H$. When discussing the dual of a graph $G$, the graph $G$ itself may be referred to as the "primal graph". Many other graph properties and structures may be translated into other natural properties and structures of the dual. For instance, cycles are dual to cuts, spanning trees are dual to the complements of spanning trees, and simple graphs (without parallel edges or self-loops) are dual to 3-edge-connected graphs.

Graph duality can help explain the structure of mazes and of drainage basins. Dual graphs have also been applied in computer vision, computational geometry, mesh generation, and the design of integrated circuits.

Cycles and dipoles
The unique planar embedding of a cycle graph divides the plane into only two regions, the inside and outside of the cycle, by the Jordan curve theorem. However, in an $n$-cycle, these two regions are separated from each other by $n$ different edges. Therefore, the dual graph of the $n$-cycle is a multigraph with two vertices (dual to the regions), connected to each other by $n$ dual edges. Such a graph is called a multiple edge, linkage, or sometimes a dipole graph. Conversely, the dual to an $n$-edge dipole graph is an $n$-cycle.

Dual polyhedra
According to Steinitz's theorem, every polyhedral graph (the graph formed by the vertices and edges of a three-dimensional convex polyhedron) must be planar and 3-vertex-connected, and every 3-vertex-connected planar graph comes from a convex polyhedron in this way. Every three-dimensional convex polyhedron has a dual polyhedron; the dual polyhedron has a vertex for every face of the original polyhedron, with two dual vertices adjacent whenever the corresponding two faces share an edge. Whenever two polyhedra are dual, their graphs are also dual. For instance the Platonic solids come in dual pairs, with the octahedron dual to the cube, the dodecahedron dual to the icosahedron, and the tetrahedron dual to itself. Polyhedron duality can also be extended to duality of higher dimensional polytopes, but this extension of geometric duality does not have clear connections to graph-theoretic duality.

Self-dual graphs
A plane graph is said to be self-dual if it is isomorphic to its dual graph. The wheel graphs provide an infinite family of self-dual graphs coming from self-dual polyhedra (the pyramids). However, there also exist self-dual graphs that are not polyhedral, such as the one shown. describe two operations, adhesion and explosion, that can be used to construct a self-dual graph containing a given planar graph; for instance, the self-dual graph shown can be constructed as the adhesion of a tetrahedron with its dual.

It follows from Euler's formula that every self-dual graph with $n$ vertices has exactly $2n − 2$ edges. Every simple self-dual planar graph contains at least four vertices of degree three, and every self-dual embedding has at least four triangular faces.

Properties
Many natural and important concepts in graph theory correspond to other equally natural but different concepts in the dual graph. Because the dual of the dual of a connected plane graph is isomorphic to the primal graph, each of these pairings is bidirectional: if concept $X$ in a planar graph corresponds to concept $Y$ in the dual graph, then concept $Y$ in a planar graph corresponds to concept $X$ in the dual.

Simple graphs versus multigraphs
The dual of a simple graph need not be simple: it may have self-loops (an edge with both endpoints at the same vertex) or multiple edges connecting the same two vertices, as was already evident in the example of dipole multigraphs being dual to cycle graphs. As a special case of the cut-cycle duality discussed below, the bridges of a planar graph $G$ are in one-to-one correspondence with the self-loops of the dual graph. For the same reason, a pair of parallel edges in a dual multigraph (that is, a length-2 cycle) corresponds to a 2-edge cutset in the primal graph (a pair of edges whose deletion disconnects the graph). Therefore, a planar graph is simple if and only if its dual has no 1- or 2-edge cutsets; that is, if it is 3-edge-connected. The simple planar graphs whose duals are simple are exactly the 3-edge-connected simple planar graphs. This class of graphs includes, but is not the same as, the class of 3-vertex-connected simple planar graphs. For instance, the figure showing a self-dual graph is 3-edge-connected (and therefore its dual is simple) but is not 3-vertex-connected.

Uniqueness
Because the dual graph depends on a particular embedding, the dual graph of a planar graph is not unique, in the sense that the same planar graph can have non-isomorphic dual graphs. In the picture, the blue graphs are isomorphic but their dual red graphs are not. The upper red dual has a vertex with degree 6 (corresponding to the outer face of the blue graph) while in the lower red graph all degrees are less than 6.

Hassler Whitney showed that if the graph is 3-connected then the embedding, and thus the dual graph, is unique. By Steinitz's theorem, these graphs are exactly the polyhedral graphs, the graphs of convex polyhedra. A planar graph is 3-vertex-connected if and only if its dual graph is 3-vertex-connected. More generally, a planar graph has a unique embedding, and therefore also a unique dual, if and only if it is a subdivision of a 3-vertex-connected planar graph (a graph formed from a 3-vertex-connected planar graph by replacing some of its edges by paths). For some planar graphs that are not 3-vertex-connected, such as the complete bipartite graph $K_{2,4}$, the embedding is not unique, but all embeddings are isomorphic. When this happens, correspondingly, all dual graphs are isomorphic.

Because different embeddings may lead to different dual graphs, testing whether one graph is a dual of another (without already knowing their embeddings) is a nontrivial algorithmic problem. For biconnected graphs, it can be solved in polynomial time by using the SPQR trees of the graphs to construct a canonical form for the equivalence relation of having a shared mutual dual. For instance, the two red graphs in the illustration are equivalent according to this relation. However, for planar graphs that are not biconnected, this relation is not an equivalence relation and the problem of testing mutual duality is NP-complete.

Cuts and cycles
A cutset in an arbitrary connected graph is a subset of edges defined from a partition of the vertices into two subsets, by including an edge in the subset when it has one endpoint on each side of the partition. Removing the edges of a cutset necessarily splits the graph into at least two connected components. A minimal cutset (also called a bond) is a cutset with the property that every proper subset of the cutset is not itself a cut. A minimal cutset of a connected graph necessarily separates its graph into exactly two components, and consists of the set of edges that have one endpoint in each component. A simple cycle is a connected subgraph in which each vertex of the cycle is incident to exactly two edges of the cycle.

In a connected planar graph $G$, every simple cycle of $G$ corresponds to a minimal cutset in the dual of $G$, and vice versa. This can be seen as a form of the Jordan curve theorem: each simple cycle separates the faces of $G$ into the faces in the interior of the cycle and the faces of the exterior of the cycle, and the duals of the cycle edges are exactly the edges that cross from the interior to the exterior. The girth of any planar graph (the size of its smallest cycle) equals the edge connectivity of its dual graph (the size of its smallest cutset).

This duality extends from individual cutsets and cycles to vector spaces defined from them. The cycle space of a graph is defined as the family of all subgraphs that have even degree at each vertex; it can be viewed as a vector space over the two-element finite field, with the symmetric difference of two sets of edges acting as the vector addition operation in the vector space. Similarly, the cut space of a graph is defined as the family of all cutsets, with vector addition defined in the same way. Then the cycle space of any planar graph and the cut space of its dual graph are isomorphic as vector spaces. Thus, the rank of a planar graph (the dimension of its cut space) equals the cyclotomic number of its dual (the dimension of its cycle space) and vice versa. A cycle basis of a graph is a set of simple cycles that form a basis of the cycle space (every even-degree subgraph can be formed in exactly one way as a symmetric difference of some of these cycles). For edge-weighted planar graphs (with sufficiently general weights that no two cycles have the same weight) the minimum-weight cycle basis of the graph is dual to the Gomory–Hu tree of the dual graph, a collection of nested cuts that together include a minimum cut separating each pair of vertices in the graph. Each cycle in the minimum weight cycle basis has a set of edges that are dual to the edges of one of the cuts in the Gomory–Hu tree. When cycle weights may be tied, the minimum-weight cycle basis may not be unique, but in this case it is still true that the Gomory–Hu tree of the dual graph corresponds to one of the minimum weight cycle bases of the graph.

In directed planar graphs, simple directed cycles are dual to directed cuts (partitions of the vertices into two subsets such that all edges go in one direction, from one subset to the other). Strongly oriented planar graphs (graphs whose underlying undirected graph is connected, and in which every edge belongs to a cycle) are dual to directed acyclic graphs in which no edge belongs to a cycle. To put this another way, the strong orientations of a connected planar graph (assignments of directions to the edges of the graph that result in a strongly connected graph) are dual to acyclic orientations (assignments of directions that produce a directed acyclic graph). In the same way, dijoins (sets of edges that include an edge from each directed cut) are dual to feedback arc sets (sets of edges that include an edge from each cycle).

Spanning trees
A spanning tree may be defined as a set of edges that, together with all of the vertices of the graph, forms a connected and acyclic subgraph. But, by cut-cycle duality, if a set $S$ of edges in a planar graph $G$ is acyclic (has no cycles), then the set of edges dual to $S$ has no cuts, from which it follows that the complementary set of dual edges (the duals of the edges that are not in $S$) forms a connected subgraph. Symmetrically, if $S$ is connected, then the edges dual to the complement of $S$ form an acyclic subgraph. Therefore, when $S$ has both properties – it is connected and acyclic – the same is true for the complementary set in the dual graph. That is, each spanning tree of $G$ is complementary to a spanning tree of the dual graph, and vice versa. Thus, the edges of any planar graph and its dual can together be partitioned (in multiple different ways) into two spanning trees, one in the primal and one in the dual, that together extend to all the vertices and faces of the graph but never cross each other. In particular, the minimum spanning tree of $G$ is complementary to the maximum spanning tree of the dual graph. However, this does not work for shortest path trees, even approximately: there exist planar graphs such that, for every pair of a spanning tree in the graph and a complementary spanning tree in the dual graph, at least one of the two trees has distances that are significantly longer than the distances in its graph.

An example of this type of decomposition into interdigitating trees can be seen in some simple types of mazes, with a single entrance and no disconnected components of its walls. In this case both the maze walls and the space between the walls take the form of a mathematical tree. If the free space of the maze is partitioned into simple cells (such as the squares of a grid) then this system of cells can be viewed as an embedding of a planar graph, in which the tree structure of the walls forms a spanning tree of the graph and the tree structure of the free space forms a spanning tree of the dual graph. Similar pairs of interdigitating trees can also be seen in the tree-shaped pattern of streams and rivers within a drainage basin and the dual tree-shaped pattern of ridgelines separating the streams.

This partition of the edges and their duals into two trees leads to a simple proof of Euler’s formula $V − E + F = 2$ for planar graphs with $V$ vertices, $E$ edges, and $F$ faces. Any spanning tree and its complementary dual spanning tree partition the edges into two subsets of $V − 1$ and $F − 1$ edges respectively, and adding the sizes of the two subsets gives the equation

which may be rearranged to form Euler's formula. According to Duncan Sommerville, this proof of Euler's formula is due to K. G. C. Von Staudt’s Geometrie der Lage (Nürnberg, 1847).

In nonplanar surface embeddings the set of dual edges complementary to a spanning tree is not a dual spanning tree. Instead this set of edges is the union of a dual spanning tree with a small set of extra edges whose number is determined by the genus of the surface on which the graph is embedded. The extra edges, in combination with paths in the spanning trees, can be used to generate the fundamental group of the surface.

Additional properties
Any counting formula involving vertices and faces that is valid for all planar graphs may be transformed by planar duality into an equivalent formula in which the roles of the vertices and faces have been swapped. Euler's formula, which is self-dual, is one example. Another given by Harary involves the handshaking lemma, according to which the sum of the degrees of the vertices of any graph equals twice the number of edges. In its dual form, this lemma states that in a plane graph, the sum of the numbers of sides of the faces of the graph equals twice the number of edges.

The medial graph of a plane graph is isomorphic to the medial graph of its dual. Two planar graphs can have isomorphic medial graphs only if they are dual to each other.

A planar graph with four or more vertices is maximal (no more edges can be added while preserving planarity) if and only if its dual graph is both 3-vertex-connected and 3-regular.

A connected planar graph is Eulerian (has even degree at every vertex) if and only if its dual graph is bipartite. A Hamiltonian cycle in a planar graph $G$ corresponds to a partition of the vertices of the dual graph into two subsets (the interior and exterior of the cycle) whose induced subgraphs are both trees. In particular, Barnette's conjecture on the Hamiltonicity of cubic bipartite polyhedral graphs is equivalent to the conjecture that every Eulerian maximal planar graph can be partitioned into two induced trees.

If a planar graph $G$ has Tutte polynomial $E = (V − 1) + (F − 1)$, then the Tutte polynomial of its dual graph is obtained by swapping $x$ and $y$. For this reason, if some particular value of the Tutte polynomial provides information about certain types of structures in $G$, then swapping the arguments to the Tutte polynomial will give the corresponding information for the dual structures. For instance, the number of strong orientations is $T_{G}(x,y)$ and the number of acyclic orientations is $T_{G}(0,2)$. For bridgeless planar graphs, graph colorings with $k$ colors correspond to nowhere-zero flows modulo $k$ on the dual graph. For instance, the four color theorem (the existence of a 4-coloring for every planar graph) can be expressed equivalently as stating that the dual of every bridgeless planar graph has a nowhere-zero 4-flow. The number of $k$-colorings is counted (up to an easily computed factor) by the Tutte polynomial value $T_{G}(2,0)$ and dually the number of nowhere-zero $k$-flows is counted by $T_{G}(1 − k,0)$.

An st-planar graph is a connected planar graph together with a bipolar orientation of that graph, an orientation that makes it acyclic with a single source and a single sink, both of which are required to be on the same face as each other. Such a graph may be made into a strongly connected graph by adding one more edge, from the sink back to the source, through the outer face. The dual of this augmented planar graph is itself the augmentation of another st-planar graph.

Directed graphs
In a directed plane graph, the dual graph may be made directed as well, by orienting each dual edge by a 90° clockwise turn from the corresponding primal edge. Strictly speaking, this construction is not a duality of directed planar graphs, because starting from a graph $G$ and taking the dual twice does not return to $G$ itself, but instead constructs a graph isomorphic to the transpose graph of $G$, the graph formed from $G$ by reversing all of its edges. Taking the dual four times returns to the original graph.

Weak dual
The weak dual of a plane graph is the subgraph of the dual graph whose vertices correspond to the bounded faces of the primal graph. A plane graph is outerplanar if and only if its weak dual is a forest. For any plane graph $G$, let $T_{G}(0,1 − k)$ be the plane multigraph formed by adding a single new vertex $v$ in the unbounded face of $G$, and connecting $v$ to each vertex of the outer face (multiple times, if a vertex appears multiple times on the boundary of the outer face); then, $G$ is the weak dual of the (plane) dual of $G^{+}$.

Infinite graphs and tessellations
The concept of duality applies as well to infinite graphs embedded in the plane as it does to finite graphs. However, care is needed to avoid topological complications such as points of the plane that are neither part of an open region disjoint from the graph nor part of an edge or vertex of the graph. When all faces are bounded regions surrounded by a cycle of the graph, an infinite planar graph embedding can also be viewed as a tessellation of the plane, a covering of the plane by closed disks (the tiles of the tessellation) whose interiors (the faces of the embedding) are disjoint open disks. Planar duality gives rise to the notion of a dual tessellation, a tessellation formed by placing a vertex at the center of each tile and connecting the centers of adjacent tiles.

The concept of a dual tessellation can also be applied to partitions of the plane into finitely many regions. It is closely related to but not quite the same as planar graph duality in this case. For instance, the Voronoi diagram of a finite set of point sites is a partition of the plane into polygons within which one site is closer than any other. The sites on the convex hull of the input give rise to unbounded Voronoi polygons, two of whose sides are infinite rays rather than finite line segments. The dual of this diagram is the Delaunay triangulation of the input, a planar graph that connects two sites by an edge whenever there exists a circle that contains those two sites and no other sites. The edges of the convex hull of the input are also edges of the Delaunay triangulation, but they correspond to rays rather than line segments of the Voronoi diagram. This duality between Voronoi diagrams and Delaunay triangulations can be turned into a duality between finite graphs in either of two ways: by adding an artificial vertex at infinity to the Voronoi diagram, to serve as the other endpoint for all of its rays, or by treating the bounded part of the Voronoi diagram as the weak dual of the Delaunay triangulation. Although the Voronoi diagram and Delaunay triangulation are dual, their embedding in the plane may have additional crossings beyond the crossings of dual pairs of edges. Each vertex of the Delaunay triangle is positioned within its corresponding face of the Voronoi diagram. Each vertex of the Voronoi diagram is positioned at the circumcenter of the corresponding triangle of the Delaunay triangulation, but this point may lie outside its triangle.

Nonplanar embeddings
The concept of duality can be extended to graph embeddings on two-dimensional manifolds other than the plane. The definition is the same: there is a dual vertex for each connected component of the complement of the graph in the manifold, and a dual edge for each graph edge connecting the two dual vertices on either side of the edge. In most applications of this concept, it is restricted to embeddings with the property that each face is a topological disk; this constraint generalizes the requirement for planar graphs that the graph be connected. With this constraint, the dual of any surface-embedded graph has a natural embedding on the same surface, such that the dual of the dual is isomorphic to and isomorphically embedded to the original graph. For instance, the complete graph $G^{+}$ is a toroidal graph: it is not planar but can be embedded in a torus, with each face of the embedding being a triangle. This embedding has the Heawood graph as its dual graph.

The same concept works equally well for non-orientable surfaces. For instance, $K_{7}$ can be embedded in the projective plane with ten triangular faces as the hemi-icosahedron, whose dual is the Petersen graph embedded as the hemi-dodecahedron.

Even planar graphs may have nonplanar embeddings, with duals derived from those embeddings that differ from their planar duals. For instance, the four Petrie polygons of a cube (hexagons formed by removing two opposite vertices of the cube) form the hexagonal faces of an embedding of the cube in a torus. The dual graph of this embedding has four vertices forming a complete graph $K_{6}$ with doubled edges. In the torus embedding of this dual graph, the six edges incident to each vertex, in cyclic order around that vertex, cycle twice through the three other vertices. In contrast to the situation in the plane, this embedding of the cube and its dual is not unique; the cube graph has several other torus embeddings, with different duals.

Many of the equivalences between primal and dual graph properties of planar graphs fail to generalize to nonplanar duals, or require additional care in their generalization.

Another operation on surface-embedded graphs is the Petrie dual, which uses the Petrie polygons of the embedding as the faces of a new embedding. Unlike the usual dual graph, it has the same vertices as the original graph, but generally lies on a different surface. Surface duality and Petrie duality are two of the six Wilson operations, and together generate the group of these operations.

Matroids and algebraic duals
An algebraic dual of a connected graph $G$ is a graph $K_{7}$ such that $G$ and $K_{6}$ have the same set of edges, any cycle of $G$ is a cut of $K_{4}$, and any cut of $G$ is a cycle of $G^{*}$. Every planar graph has an algebraic dual, which is in general not unique (any dual defined by a plane embedding will do). The converse is actually true, as settled by Hassler Whitney in Whitney's planarity criterion:

 * A connected graph $G$ is planar if and only if it has an algebraic dual.

The same fact can be expressed in the theory of matroids. If $M$ is the graphic matroid of a graph $G$, then a graph $G^{*}$ is an algebraic dual of $G$ if and only if the graphic matroid of $G^{*}$ is the dual matroid of $M$. Then Whitney's planarity criterion can be rephrased as stating that the dual matroid of a graphic matroid $M$ is itself a graphic matroid if and only if the underlying graph $G$ of $M$ is planar. If $G$ is planar, the dual matroid is the graphic matroid of the dual graph of $G$. In particular, all dual graphs, for all the different planar embeddings of $G$, have isomorphic graphic matroids.

For nonplanar surface embeddings, unlike planar duals, the dual graph is not generally an algebraic dual of the primal graph. And for a non-planar graph $G$, the dual matroid of the graphic matroid of $G$ is not itself a graphic matroid. However, it is still a matroid whose circuits correspond to the cuts in $G$, and in this sense can be thought of as a combinatorially generalized algebraic dual of $G$.

The duality between Eulerian and bipartite planar graphs can be extended to binary matroids (which include the graphic matroids derived from planar graphs): a binary matroid is Eulerian if and only if its dual matroid is bipartite.

The two dual concepts of girth and edge connectivity are unified in matroid theory by matroid girth: the girth of the graphic matroid of a planar graph is the same as the graph's girth, and the girth of the dual matroid (the graphic matroid of the dual graph) is the edge connectivity of the graph.

Applications
Along with its use in graph theory, the duality of planar graphs has applications in several other areas of mathematical and computational study.

In geographic information systems, flow networks (such as the networks showing how water flows in a system of streams and rivers) are dual to cellular networks describing drainage divides. This duality can be explained by modeling the flow network as a spanning tree on a grid graph of an appropriate scale, and modeling the drainage divide as the complementary spanning tree of ridgelines on the dual grid graph.

In computer vision, digital images are partitioned into small square pixels, each of which has its own color. The dual graph of this subdivision into squares has a vertex per pixel and an edge between pairs of pixels that share an edge; it is useful for applications including clustering of pixels into connected regions of similar colors.

In computational geometry, the duality between Voronoi diagrams and Delaunay triangulations implies that any algorithm for constructing a Voronoi diagram can be immediately converted into an algorithm for the Delaunay triangulation, and vice versa. The same duality can also be used in finite element mesh generation. Lloyd's algorithm, a method based on Voronoi diagrams for moving a set of points on a surface to more evenly spaced positions, is commonly used as a way to smooth a finite element mesh described by the dual Delaunay triangulation. This method improves the mesh by making its triangles more uniformly sized and shaped.

In the synthesis of CMOS circuits, the function to be synthesized is represented as a formula in Boolean algebra. Then this formula is translated into two series–parallel multigraphs. These graphs can be interpreted as circuit diagrams in which the edges of the graphs represent transistors, gated by the inputs to the function. One circuit computes the function itself, and the other computes its complement. One of the two circuits is derived by converting the conjunctions and disjunctions of the formula into series and parallel compositions of graphs, respectively. The other circuit reverses this construction, converting the conjunctions and disjunctions of the formula into parallel and series compositions of graphs. These two circuits, augmented by an additional edge connecting the input of each circuit to its output, are planar dual graphs.

History
The duality of convex polyhedra was recognized by Johannes Kepler in his 1619 book Harmonices Mundi. Recognizable planar dual graphs, outside the context of polyhedra, appeared as early as 1725, in Pierre Varignon's posthumously published work, Nouvelle Méchanique ou Statique. This was even before Leonhard Euler's 1736 work on the Seven Bridges of Königsberg that is often taken to be the first work on graph theory. Varignon analyzed the forces on static systems of struts by drawing a graph dual to the struts, with edge lengths proportional to the forces on the struts; this dual graph is a type of Cremona diagram. In connection with the four color theorem, the dual graphs of maps (subdivisions of the plane into regions) were mentioned by Alfred Kempe in 1879, and extended to maps on non-planar surfaces by in 1891. Duality as an operation on abstract planar graphs was introduced by Hassler Whitney in 1931.

title: Dual graph


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The Fudge Family

The Fudge Family may refer to two works by the Irish-born writer Thomas Moore.

 * The Fudge Family in Paris (1818)
 * The Fudge Family in England (1835)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: The Fudge Family


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The Fudge Family in Paris

The Fudge Family in Paris is an 1818 verse satire by Thomas Moore. It was intended to be a comedic critique of the post-war settlement of Europe following the Congress of Vienna and of the large number of British and Irish families who flocked to France for tourism. Written after a brief trip to Paris, its popular success inspired several more parodies and replies.

The work
The Fudge Family in Paris was written following Thomas Moore's visit to Paris in company with Samuel Rogers in 1817. Such was its popularity after its appearance on 10 April 1818 from the Longman partnership that five editions were published in its first fortnight, with a further four editions that year as well as American editions from New York and Philadelphia. On its title page the work's editorship was attributed to Thomas Brown the Younger (alluding to the satirist of a century before), under which name Moore had published earlier epistolary political satires under the title of Intercepted Letters (1813). However, the true authorship of the Fudge correspondence was soon identified.

Ostensibly the work is a collection of letters that depict the visit to Paris of the Fudge family in the wake of the Bourbon Restoration. The party is made up of four characters who record their very different impressions. The father, Phil Fudge, is researching a book which he intends to be propaganda on behalf of his patron, Lord Castlereagh, whose parliamentary speeches he parrots. His son Bob is a dandy and glutton, while his daughter Biddy is in search of romance and falls in love with a commoner whom she believes to be royalty in disguise. Their tutor, Phelim Connor, on the other hand, is an Irish revolutionary and Bonapartist opposed to everything that his employer (and distant relative) stands for. Each character is further differentiated by the poetic form of their letters: Phelim declaims in decasyllables, Phil trips in lighter octosyllables, while Bob and Biddy chatter colloquially in anapaestic measure.

As well as a comic social satire, the poem is also an attack on the tyrannical politics of the Holy Alliance and English policy in Ireland, identified with the British Tory party and particularly the foreign secretary, Lord Castlereagh. It was the political themes that William Hazlitt preferred in his contemporary review of The Fudge Family in Paris, soon collected in his Political Essays (1819), making of it an opportunity to bludgeon such Lake Poet turncoats as Southey, Wordsworth and Coleridge. Another critic allowed that the work was for its time "a happy blending of the political squib and the social burlesque", yet over the decades "it is the natural fate of ephemeral satire to perish with the events which gave rise to it".

Literary historians in the following century saw in Moore's poem a successful and influential imitation of Christopher Anstey's The New Bath Guide and in the line of the "anapaestic travel narratives" that Anstey had inspired. There has, however, been uncertainty over defining its literary genre. The work has been described as a "satirical travelogue", a "verse satire in epistolary form", and even as a precursor of the Victorian "verse novel".

Imitations and sequels
In the same year that The Fudge Family in Paris was launched, it was followed by the imitative Replies to the Letters of the Fudge Family in Paris, using the same epistolary form and anapaestic measure and pretending to be "edited by Thomas Brown Esq." Moore was quick to disavow the work that many were taking for his as a "catchpenny" derivative and "an impudent thing". It was not long, either, before Moore came under attack from the satirised Tory side in The Fudger Fudged (London 1819).

Moore's work was followed by several other imitations in the next few years. They included The Fudge Family in Edinburgh in a series of Poetical Epistles, ascribed to a pseudonymous Nehemiah Nettlebottom (Edinburgh 1820). This centred on the Caledonian tour of Bob Fudge and had a complicated plot recounted over 19 letters and involving seven characters. The contemporary The Fudge Family in Washington, supposed to have been "edited by Harry Nimrod" (Baltimore 1820), introduced a Fudge family relative from Virginia on a trip to the American capital. This included both political satire and comedy, as represented by the contrasting prudish and permissive social attitudes of two female visitors. Two years later came Fudge in Ireland (London 1822), a hybrid work that has been ascribed to William Russell Macdonald (1787-1854) and describing itself as "a collection of letters, poems and legends, concerning the castle, the courts, the college, the corporation and the country at large".

Longman had originally proposed the writing of a sequel, The Fudge Family in London, but before Moore got round to it the subject was taken up in part by another imitator. In two sections of this, The Fudge Family in England (London 1823), another new character is introduced in the shape of a French Fudge who describes his stay in the English capital to his English relative in Paris. While Moore was living in financial exile abroad at this period, he had begun writing instead The Fudge Family in Italy but had to abandon the project. It was not until 1835 that he finally wrote his own variation on the theme, The Fudges in England, set in a spa-town reminiscent of the scene of Anstey's New Bath Guide.

title: The Fudge Family in Paris


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Talk:Devil's Pitchfork

This is an attempted redirect to Impossible objects; in fact it should be a redirect to Blivet, an "impossible object" also known as a Devil's pitchfork. I've set up the attempted redirect properly Tonywalton | Talk 16:37, 25 August 2005 (UTC)

title: Talk:Devil's Pitchfork


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Wikipedia:Peer review/Sonic the Hedgehog (character)/archive1

Sonic the Hedgehog (character)
Since the CVG Peer Review wasn't very helping (and Mario is also aching for the FA), I need to know what is needed here. igordebraga ≠ 16:38, 25 August 2005 (UTC)

 * It's a very decent start, but I think it still needs some work.
 * Statements such as "incredibly popular", "overwhelming popularity" and "Sonic is well known for his various abilities" are non-encyclopedic.
 * In some places it is unclear what is canon, what is non-canon, and from whence the non-canon material was derived. An example is the "Former US/UK version" section, where it expounds on the coloration question that was earlier stated as non-canon. But what else in that section is non-canon?
 * There's also an issue with the chronology in places. For example is says "In more recent games...", but never explains what is meant by "recent". How will that read in 20 years? Likewise for "In later games he...". Specific dates would be better.
 * It could use a reference section, instead of general statements like "American sources often claim" or "by some sources, 16-year-old".
 * Is there any information on who is credited with the creation of this character, besides the company?
 * I hope this was helpful. Thanks. :) — RJH 17:33, 26 August 2005 (UTC)

 * A script has been used to generate a semi-automated review of the article for issues relating to grammar and house style. If you would find such a review helpful, please click here. Thanks, APR t 03:39, 18 January 2008 (UTC)


text: User talk:205.211.160.1

title: User talk:205.211.160.1


text:
Talk:Hooghly district

Requested move
Move to Hooghly district as more in line with other districts; can't move due to redirect here - Master Of Ninja 16:39, 25 August 2005 (UTC)

 * Add *Support or *Oppose followed by an optional one sentence explanation, then sign your vote with ~

Discussion

 * Add any additional comments

"Hooghly district" may be consistent, but I think it would be incorrect; "Hooghly District" is a proper noun, and as such should be capitalized. Tom Radulovich 16:12, 26 August 2005 (UTC)

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text:
Crawford Auto-Aviation Museum

The Crawford Auto-Aviation Museum is a transportation museum in Cleveland, Ohio, United States. It is part of the Western Reserve Historical Society's Cleveland History Center in University Circle, and its collection includes about 170 cars. It was founded by Frederick C. Crawford of TRW, and opened in 1965.

Collections
As of 2019 there were more than 170 automobiles, 12 aircraft, 3 antique carriages, and 21 non-car artifacts (motorcycles, boats and bicycles). The facility includes more than 2000 sqft of archival collections.

The aviation collection includes a P-51 Mustang racing plane used in Thompson Trophy Races. The oldest car in the collection is an 1897 Panhard et Levassor, while later acquisitions include the first production DeLorean from 1981 and a self-driving car named DEXTER which was team Team Case's entry in the DARPA Urban Challenge 2007, in which it placed in the top 20.

Other vehicles in the collection include Tinkerbelle, a small sailboat in which Robert Manry solo sailed across the Atlantic Ocean in 1965. Manry donated it to the Historical Society in 1967.

In 2016, the museum's 1913 ALCO Model Six Berline Limousine won the Ansel Adams Award at the Pebble Beach Concours d'Elegance in California.

History
The collection began as the Thompson Products Auto Album, which was also founded by Crawford. Crawford explained that when he started collecting the cars, it was simply because it seemed a shame to let them be scrapped, which was the typical fate of almost all antique machinery at the time. He saw value in saving a few historically significant examples.

In 1990, the museum sold off almost 70 automobiles by putting them up for auction with Sotheby's.

To pay down debt, the museum sold or auctioned 44 cars in 2009, 24 of them through RM Auctions in October. The deaccessions proceeded over public protest and the objections of Kay Crawford, the widow of founder Frederick C. Crawford. The museum also sold a Goodyear F2G Corsair it had purchased from Walter Soplata and a Airco DH.4 originally acquired by Crawford.

As of 2018, the museum featured two major exhibits: Setting the World in Motion, featuring cars and airplanes made in Northeast Ohio, and REVolution: The Automobile in America, telling the story of the automobile in America.

The museum dismissed its director, Brad Brownell, in 2023 due to disagreements over object use philosophies.

title: Crawford Auto-Aviation Museum


text:
File talk:Death of Jason Todd.jpg

Um...This made me cry. 70.59.187.173

title: File talk:Death of Jason Todd.jpg


text:
User:MicroBio Hawk

DVDs I own and like to edit the pages of:
 * About A Boy
 * Adaptation.
 * Almost Famous
 * The American President
 * Airplane!
 * The Apartment
 * Amelie
 * As Good as It Gets
 * Back To The Future trilogy
 * Batman Begins
 * The Big Sleep
 * Blazing Saddles
 * Blues Brothers
 * Boondock Saints
 * Breakfast at Tiffany's
 * Casablanca
 * Catch Me If You Can
 * Charade /His Girl Friday Combo
 * Chasing Amy
 * Chicago
 * Collateral
 * A Christmas Story
 * The Daily Show: Indecision 2004 3-disc DVD set
 * Donnie Darko
 * Election
 * Enemy at the Gates
 * Ferris Bueller’s Day Off
 * Fight Club
 * First Strike (Jackie Chan)
 * A Fish Called Wanda
 * Forrest Gump
 * The Fugitive
 * Gattaca
 * Get Shorty
 * Glengarry Glen Ross
 * Good Morning, Vietnam
 * Good Will Hunting
 * The Great Escape
 * The Green Mile
 * Groundhog Day
 * Grumpy Old Men
 * High Fidelity
 * Hot Shots
 * IQ
 * Indiana Jones and the Raiders of the Lost Ark
 * Indiana Jones and the Temple of Doom
 * Indiana Jones and the Last Crusade
 * Indiana Jones Bonus Materials
 * The Insider
 * Insomnia
 * The Italian Job
 * Jackass: The Movie
 * Jay and Silent Bob Strike Back
 * Jerry Maguire
 * Kill Bill, Vol. 1
 * Kill Bill, Vol. 2
 * L.A. Confidential
 * A League of Their Own
 * Legend of Drunken Master (Jackie Chan)
 * Life Is Beautiful
 * Lord of the Rings: Fellowship of the Ring (4-disc DVD set)
 * Lord of the Rings: The Two Towers (4-disc DVD set)
 * Lord of the Rings: Return of the King (4-disc DVD set)
 * M.A.S.H.
 * The Mask of Zorro
 * The Matrix
 * The Matrix Reloaded
 * Memento
 * Minority Report
 * Monty Python and the Holy Grail
 * Mr. Nice Guy (Jackie Chan)
 * My Cousin Vinny
 * Network (film)
 * North By Northwest
 * Office Space
 * Ocean’s Eleven
 * One Flew Over the Cuckoo’s Nest
 * Phenomenon
 * The Princess Bride
 * The Producers
 * Pulp Fiction: Special Edition
 * Quiz Show
 * Requiem for a Dream
 * Reservoir Dogs
 * Roman Holiday
 * Roxanne
 * Rumble In the Bronx (Jackie Chan)
 * Sabrina
 * The Score
 * Shanghai Knights
 * Shanghai Noon
 * The Shawshank Redemption
 * Signs
 * Sin City
 * Singin’ In the Rain
 * The Sixth Sense
 * Sneakers
 * Some Like It Hot
 * Spartan
 * Spider-Man
 * Spider-Man 2
 * Stalag 17
 * Star Wars: Original trilogy (4-disc DVD set)
 * Star Wars: Episode II
 * Stripes
 * There's Something About Mary
 * This Is Spinal Tap
 * The Thomas Crown Affair
 * The Truman Show
 * True Lies
 * The Truth About Cats & Dogs []
 * Young Frankenstein []
 * X2

title: User:MicroBio Hawk


text:
Jean-Jacques Boisard

Jean-Jacques François Marius Boisard (Caen, 1743 – 1831) was a French fabulist.

Biography
Boisard was a French fabulist born in 1743 in Caen, a historical town located in Normandy, North-West France, about 150 kilometers from Paris. Educated by his family to be a member of the local Bar, he was appointed Secretary of the Finance Council and, in 1778, Secretary of the Chancery of "Monsieur", namely the elder brother of the king of France. With the French Revolution, he lost his post and, later on, the modest pension allocated to it.

As a member of the Royal Academy of Literature (Académie royale des Belles-lettres) of Caen, the first academy established in France after the "Académie française", he started publishing fables in 1764 in the Mercure de France. Fables were very popular in the French literature of the 18th century, the Age of Enlightenment, under the shadow of Jean de La Fontaine. In 1771, the publisher Lottin published the Fablier français, which was a compilation of texts from some hundred authors who wrote fables, since La Fontaine. This literary art dates back, at least, to Greek antiquity. Fables were usually written in verses and used animals for their symbolic value. They carry implicit or explicit messages of moral philosophy.

Facing problems due to his antirevolutionary ideas against the Republic, Boisard left Paris and returned to Caen. The city was incidentally a centre of "federalist" insurgents against the revolutionary government. One of its citizens, Charlotte Corday, in 1793, murdered a prominent leader of the Revolution, Jean Paul Marat. Boisard was, at this time, apparently not involved in politics. Up to 1805, he published several collections of fables and various works. The richness of his imagination was matchless.

At the time, he was recognized as a very respectable author. Beside the fables, he wrote some light poetry and translated several poems of Horace. Voltaire praised him, Diderot (Epitre à Boisard) dedicated one of his very few rhymed Epistle to the fabulist: "nobody has been such a charming rhyme writer ... Nobody has been such a perfect rhyme writer" ("Oncques ne fut un rimeur si charmant ... Oncques ne fut un rimeur si parfait"). It must be noticed however that modern literary critics are now of the opinion that this epistle may have been addressed to his nephew (mentioned below) since it implicitly refers to a bad fortune. For his part, Jacob Grimm, who was a close friend, considered Boisard's fables as natural and diverse, but sometimes naïve. He was quite laudatory in writing that Boisard was the fabulist who imitated the less La Fontaine, but came the closest to the master. The polemical writer Antoine de Rivarol was sarcastic in stating that Boisard's fables "have outmoded those of La Fontaine, which is always somewhat unfair" ("ont fait passer de mode celles de La Fontaine, ce qui est toujours un peu injuste!")

Boisard is one of the most prolific authors of French literature. His publications were produced over a period of 33 years (1773 to 1806). Presently, he is broadly considered as the best fabulist after La Fontaine, although his literary style, pleasant and light but sometimes inelegant, is certainly not of the same standard. In 1806, Mille et une fables (A Thousand and One Fables) was published as a compilation of at least five former editions. It was a bulky and sumptuously printed and illustrated publication, which does not seem to be available now. As with hundreds of French fabulists of that time, except La Fontaine, the work of Boisard did not really survive as centuries passed. Boisard also produced a lyrical poem, "Ode sur le déluge", which was crowned, in 1790, by the Academy of Rouen. Boisard died in his home town of Caen in 1831 or 1833.

Works
First collection of Fables published in 1773, second collection in 1777, third collection in 1803, fourth in 1804, fifth in 1805. Final publication: Mille et une fables, in 1806

title: Jean-Jacques Boisard


text:
Cliff Holton

Clifford Charles Holton (29 April 1929 – 31 May 1996) was an English footballer.

Born in Oxford, Holton played as a full back for non-league Oxford City as a youth, before joining Arsenal in October 1947 at the age of 18. He spent three seasons in Arsenal's' reserve team, eventually converting to a centre forward, although his progress was restricted in part by national service. He made his debut against Stoke City on 26 December 1950; initially a bit-part player, he became an established regular in the side in 1951–52 and remained there for three seasons. He was in the Arsenal side to reach the FA Cup Final in 1952.

With 22 goals (19 of them in the league) he helped Arsenal to a First Division title in 1952–53. However, he lost his place in the Arsenal side in 1954–55, before returning the following season (1955–56), having been converted into a utility man – playing in defence, midfield or up front. He continued to figure for Arsenal for another three seasons and on 6 October 1956, he scored four goals in a home match against Manchester City, becoming the first player since Jack Lambert to score more than three in a game at Highbury. In all he scored 88 goals in 217 matches for Arsenal. Although he was often tipped for an England place, he never won a cap for his country.

Holton was sold to Watford in October 1958 for £9,000, a somewhat surprising move, considering Watford were in the newly formed Fourth Division at the time. Holton spent three seasons with the Hornets, mostly at centre forward. He formed a successful partnership with Dennis Uphill, scoring a club record 48 goals in 1959–60, which also saw the club promoted to the Third Division. Holton was then controversially sold to Northampton Town in 1961 where he scored a club record 36 goals in 1961–62, thus achieving the rare feat of holding the all-time goals in a season record at two different clubs.

Holton went on to play for Crystal Palace, signing in December 1962 and was part of the Palace side which achieved promotion to the second tier in 1964, with 43 appearances that season scoring 20 goals. He then signed for Watford (for a second time) on 6 May 1965, before moving on to Charlton Athletic (in a deal which saw Stewart Scullion move in the opposite direction) and subsequently Leyton Orient. He finally retired in 1968, due in part to a knee injury, and left the game completely to take up a career in engineering. He died suddenly while on holiday in 1996, at the age of 67.

Honours

 * First Division: 1952–53
 * FA Charity Shield: 1953

title: Cliff Holton


text:
Talk:Cograph

Too technical
I'm not sure it's needed to explain every technical thing in the Wikipedia in a language that anyone would understand. For instance, the explanation for this still-simple article seams quite absurd (Mariano 16:46, 25 August 2005 (UTC)):

1. Can be constructed from (1) isolated vertices by (2) complement and (3) disjoint union. This means that: And, tat any Cograph can be constructed following this 3 rules.
 * 1) A group of isolated vertices is a Cograph
 * 2) if G is a Cograph, then changing the edges so that x will be adjacent to y if it was not adjacent of it in the original, but x will not be adjacent to y if it was in the original.
 * 3) if G and J are a Cograph, then the graph of putting together G and J without new edges is also a Cograph

2. Can be (1) decomposed by (2) isolated vertex elimination and complement. G ∪ {x} (G union an element x) is also a Cograph This means that we can play "She loves me, she loves me not" with the graph, taking vertices that are not connected to any other vertex, or complement the graph, and we will end up with an empty graph.

3. It contains no induced [Glossary_of_graph_theory#Path|path]] of length at least 4. A P4 of a graph is a path of 4 distinct vertices a, b, c and d, with a connected to b, b connected to c, and c connected to d, such that to go from a to b I have no shortcut. Therefore, if our graph G has a P4, then is not a Cograph, and if it doesn't have it, it is a Cograph.

4. The maximum distance between two vertices in the same connected component is at most 2. If there is a path from a to b, then the shortest path between them includes one more vertex, or none at all.

5. Has at least one pair of false or true twins (that have the same opened or closed neighbourhood). This means that if G is a Cograph, then there must exist vertices a and b such that all the neighbours of a are also neighbours of b. Bear in mind that it doesn't matter if a and b are neighbours.

Cleaning
I tried to simplify the entry. I stressed the constructive definition, which is not so difficult to handle and gave some of the possible other ones as characterizations (formally, we may not have more than one definition). I removed the characterization by twins which has mainly a technical interest, but I kept:
 * 1) the characterization by forbidden $$P_4$$
 * 2) * for historical reasons (it is one of the independent introductions of the class)
 * 3) * as it is quite simple although not obviously related to the main definition
 * 4) the characterization by the diameters of the connected subgraphs
 * 5) * as it is clearly related to the previous one
 * 6) * as it enhances the intuition of what a cograph looks like
 * 7) the characterization by clique-independent set intersection
 * 8) * as it is one the strong properties of cographs, even stronger than the property to be trivially perfect. It is the fundamental reason of the low complexity of coloring problems in this class.

I also have added some references.

pom 14:50, 14 November 2005 (UTC)

Category
As noticed by D. Eppstein, Cographs are not a parametrized family hence should not belong to Category:Graph. pom 23:56, 5 October 2006 (UTC)

P4 notation
First, my substantive comment. I augmented the first characterization by explicitly including the notation P4. Since the article's introduction points out that the cographs are also known as the P4-free graphs, I figured it would add to the article's clarity if this P4 thing showed up somewhere below to tie things together.

Now a confession. I did it once and accidentally saved the page without having cretaed a comment to document the change in the history. So I immediately undid the change and re-implemented it, this time including the comment. That explains the 3 mods in such quick succession.—PaulTanenbaum (talk) 20:28, 9 July 2008 (UTC)

Proof for "Cographs are exaktly the graphs of clique-width at most 2"
Where can I find a proof for that. I searched for that in all the given References. —Preceding unsigned comment added by 141.20.6.79 (talk) 18:38, 9 March 2011 (UTC)
 * I added a source from the clique-width article. —David Eppstein (talk) 01:22, 10 March 2011 (UTC)

 * Thanks. And now I also found the proof that a graph who does not have the path of 3 edges as induced subgraph is a cograph: see Corneil, Lerchs, Stewart Burlingham: Complement reducible graphs (Discrete Applied Mathematics 3 (1981) pp. 167+168) and Seinsche: On a Property of the Class of n-Colorable Graphs (JOURNAL OF COMBINATORIAL THEORY (B) 16, pp. 191+192 (1974)). I remarked that in the article.Faust17 (talk)

Union of graphs undefined
The concept of (disjoint) union is generally defined only for sets, which graphs are not because they are pairs of sets.

If an abuse of notation is being adopted, it should be clarified somewhere. — Preceding unsigned comment added by 213.46.252.136 (talk) 10:41, 12 January 2015 (UTC)
 * The disjoint union of graphs is standard and well defined; see Graph operations. It is a different operation than the disjoint union of sets, because it has arguments of a different type (graphs not sets). It is not an abuse of notation. —David Eppstein (talk) 17:12, 12 January 2015 (UTC)

Assessment comment
Substituted at 01:53, 5 May 2016 (UTC)

title: Talk:Cograph


text:
Pap of Armenia

Pap (c. 353 – 374/375) was king of Armenia from 370 until 374/375, and a member of the Arsacid dynasty. His reign saw a short, but notable period of stabilization after years of political turmoil. Although Armenia had been conquered and devastated by the Sassanid king Shapur II in 367/368, Pap was restored to the throne at a young age with Roman assistance in 370. Early in his reign, Armenia and Rome won a joint victory over the Persians at the Battle of Bagavan, and some former territories of the kingdom were reconquered by the efforts of his (general-in-chief) Mushegh Mamikonian. Although Pap's reign began with a reconciliation of the monarchy, nobility and church, his relations with the church soon deteriorated. Pap allegedly had the Patriarch of Armenia, Nerses I, poisoned, although some later historians doubt this narrative. Pap also eventually ran afoul of the Romans, who suspected him of colluding with the Persians. The emperor Valens unsuccessfully attempted to assassinate him in 373/374, but ultimately succeeded in having him killed in 374/375. He was succeeded by his nephew Varazdat as king.

Pap is depicted with hostility by the classical Armenian historians, likely due to the king's troublesome relationship with the Armenian Church. The Roman historian Ammianus Marcellinus, however, praised Pap for his bravery and cleverness. Some later Armenian historians have reevaluated Pap positively, valuing his attempts to strengthen the Armenian monarchy and pursue an independent foreign policy under difficult circumstances.

Family
Pap was the son of the Arsacid king of Armenia Arshak (Arsaces) II ((r. 350 – 368)) and his wife Parandzem. The exact year of Pap's birth is not known for certain and has been debated by historians; one source gives it as approximately 353. Armenian historian Hakob Manandian considered it possible that Pap was actually the son of Parandzem by her first husband Gnel (Arshak's nephew). Historian Albert Stepanyan argues that Pap was in fact Arshak's son, but that he was initially legally regarded as Gnel's son, as Arshak had married Paradzem in an Iranian-style levirate marriage called or, whereby a childless widow would marry one of her late husband's agnatic relatives to provide him with an heir. Additionally, Arshak had apparently married Parandzem while still married to his first wife Olympias, despite the recent banning of polygamy at the church council of Ashtishat. For these reasons, Arshak faced serious obstacles in legitimizing Pap as his legal son and heir. According to Stepanyan, it was only after the death of Queen Olympias (purportedly by poisoning on Parandzem's orders) that Parandzem was made a full royal consort and her son Pap was recognized as crown prince of Armenia. Historian Nina Garsoïan, on the other hand, contends that Pap must have been Arshak's legitimate son and heir, as even the sources extremely hostile to him never question his legitimacy. She proposes another hypothesis according to which Pap was born to Parandzem by Arshak around 350, after which Parandzem was passed to Gnel in a temporary marriage and taken back later by Arshak, thus explaining how Pap could have been born prior to Gnel's death circa 359.

Pap also appears to have had a brother or half-brother whose son Varazdat succeeded Pap as king. This brother is not mentioned by name in the histories of Faustus of Byzantium or Movses Khorenatsi, two of the main Armenian sources on Pap's life, but another Armenian work, the anonymous Vita of St. Nerses, reports that Pap had a younger brother named Trdat.

Pap's name is of Middle Iranian origin and literally means "father". The manuscripts of Ammianus Marcellinus's history give his name as Para, which historians read as Papa.

Ascendance to the throne
Around 367/368, Pap's father Arshak II went to Persia for peace negotiations with the Sassanid king Shapur II and was imprisoned, leaving the Armenian throne vacant. Queen Parandzem and Prince Pap took refuge with the royal treasure in the fortress of Artogerassa (Artagers), defended by a troop of (lesser Armenian nobles). According to Ammianus Marcellinus, the Persian invasion force was commanded by two Armenian defectors, Cylaces (Glak) and Artabanes (Artavan or Vahan). Faustus also mentions two Armenian (magnates), Meruzhan Artsruni and Vahan Mamikonian (possibly identifiable with Ammianus's Artabanes), in leadership positions under Shapur II's suzerainty, as well as Zik and Karen who carried Persian noble titles. Shapur II may have intended to combine Sassanid administrative rule (Zik and Karen) with that of rule (Artsruni and Mamikonian).

During the siege, Arshak II's wife Parandzem appealed to Cylaces and Artabanes in the name of her husband. The two men defected back to the Arsacid monarchy and engineered the escape of Pap. Themistius reported Pap's arrival at Valens' court in Marcianopolis, where the emperor was wintering. According to Faustus, Pap was in contact with his mother while in Roman territory and encouraged her to await his return. Valens sent him to stay at Neocaesarea in Pontus Polemoniacus, 300 km from the Armenian border, where Pap received "liberal support and education." In 369, at the request of Mushegh Mamikonian (according to Faustus) or of Cylaces and Artabanes (according to Ammianus), Valens allowed Pap to return to Armenian territory. He was accompanied by the comes et dux Terentius but was not yet recognized as King of Armenia by the Romans.

King of Armenia
Valens was reluctant to bestow a royal title upon Pap as this would violate an earlier treaty signed by Jovian in July 363, whereby Rome had pledged not to intervene in Armenian affairs. Nevertheless, Shapur was enraged at Pap's restoration and personally invaded Armenia in response, forcing Pap to leave Armenia again and go into hiding near the Roman frontier in Lazica. Instead of going after Pap, Shapur II concentrated his attack on Artogerassa, which fell in the winter of 369/70. The royal treasure was captured by the Persians and Parandzem was raped and murdered. Shapur II also began systematically persecuting the local Christians, destroying churches, erecting fire temples and forcing conversion to Zoroastrianism.

Shapur II contacted Pap while he was in hiding and tried to persuade him to come over to his side. Under Shapur II's influence, Pap murdered the duplicitous Cylaces and Artabanes and sent their heads to the shahanshah as a sign of loyalty. Shapur's attempted rapprochement with Pap was aborted, however, by Arinthaeus's return to Armenia with a Roman army that restored Pap to the throne for a second time in approximately spring 370. In the spring of 371, Shapur II launched another massive invasion of Armenia. In response, Valens sent a force headed by his generals Traianus and Vadomarius into Armenia. The Armenian army also assembled under the command of Mushegh Mamikonian. The joint Roman-Armenian force met the invading Persian army in the region of Bagrevand and emerged victorious at the Battle of Bagavan. Faustus of Byzantium gives considerable credit for the victory to Mushegh Mamikonian and writes that Pap observed the battle from the nearby height of Mount Npat. During the ensuing battles, more Armenian territories were reclaimed from the Persians (according to Faustus), including Arzanene and Corduene, which had been ceded to the Persians by Jovian in 363. By the end of the summer, Shapur II had retreated to his capital at Ctesiphon and Valens had returned to Antioch. Shapur II was unable to confront the massive Roman buildup in Armenia as a result of his preoccupation with Kushan attacks in the eastern part of his empire. Thus, Roman control over Armenia through the client king Pap was secure for the time being.

Pap was a young man—likely still a teenager—when he took the throne. At the beginning of his reign, he invited Patriarch Nerses I to return to Armenia. Nerses agreed and undertook the restoration of Armenia's churches and church institutions, caring for the poor and reestablishing the church's influence in the country. Meanwhile, Mushegh campaigned to restore Arsacid authority in Armenia, brutally punishing the provinces that had revolted against the monarchy, forcing the pro-Persian  to submit to royal authority, and retaking territories from neighboring Albania and Iberia. Soon after these initial successes, Pap came into conflict with Patriarch Nerses. According to Faustus, Nerses constantly reprimanded Pap for his sinful behavior and refused to allow him to enter the church; Movses Khorenatsi implies that Pap was upset at Nerses for having him return lands that had been confiscated from the nobility during his father's reign. Some modern historians believe that Pap clashed with the church due to his support for Valens's pro-Arian religious policy; others believe that Pap was a Christian in name only and that he was sympathetic towards Zoroastrianism. Still others regard Pap's conflict with the clergy as the result of his steps to restrain the excessive power of the church, which had accumulated significant estates and wealth in the form of the charitable institutions created by Nerses during Arshak II's reign. The conflict between the king and the patriarch came to a head in 373, when, according to Faustus and Movses, Pap invited Nerses to dinner at his mansion in the village of Khakh and had him poisoned. The king then dissolved the charitable institutions established by Nerses, abolished the and  tithes paid to the church, and seized much of the church's lands. He also limited the clergy's exemption from military service, ordering that the sons and brothers of the lower clergy be placed at the disposal of the court. Pap nominated a man named Yusik as a replacement and sent him for consecration in Caesarea, but the Bishop of Caesarea Basil refused to consecrate the nominee. Valens requested that Basil quickly resolve the situation by finding a new nominee acceptable to Pap. Basil failed to do so and the Roman see of Caesarea effectively lost its traditional role of consecrating the Patriarch of Armenia. According to Faustus, the poisoning of the popular and powerful patriarch caused a rift between the king and the nobility and alienated Mushegh in particular. Nerses had also been a close Roman contact; his murder and the subsequent loss of Roman ecclesiastical control over the appointment of the Patriarch of Armenia must have damaged Pap's relations with Valens.

Some later historians have cast doubt on or totally rejected the assertion that Pap had Nerses poisoned. The Armenian historian Leo considered it a legend that was presented as fact by later ecclesiastical historians seeking to defame Pap. Authors Malachia Ormanian and Yeghiazar Muradian, judging from the circumstances described by Faustus, thought it more likely that Nerses died of some illness of the lungs or heart, perhaps on the same day or the day after he had dinner with Pap, giving rise to the rumor that the patriarch had been poisoned. Josef Markwart and Hakob Manandian also reject the story of Nerses's poisoning, arguing that Pap would have surely been called to account for it by Basil of Caesarea. Nina Garsoïan and Noel Lenski, on the other hand, do not dispute that Pap poisoned Nerses and note that he was not the first Arsacid king to have a patriach murdered, in reference to Pap's grandfather Tiran who had Patriarch Yusik assassinated. Ammianus Marcellinus is notably silent on Nerses's murder. This may reflect the Roman historian's general disinterest in religious matters, although it has been suggested that Ammianus deliberately omitted this episode in order not to diminish his narrative of Pap as "the innocent victim of Roman villainy."

Fall
In addition to the controversy over the appointment of a new patriarch, Pap's relations with Valens further suffered due to the Roman commander Terentius, who wrote to the emperor criticizing Pap and advising him to depose the Armenian king in order to prevent him from defecting to Persia. According to Faustus, Pap also demanded control over Caesarea and twelve other Roman cities including Edessa as former Arsacid domains while openly courting Persia, in defiance of the warnings of Mushegh and other nobles not to break the alliance with Rome. Ammianus, on the contrary, claims that Pap was completely loyal to Rome. Valens decided to have Pap executed, and invited him to a meeting in Tarsus in 373 or 374. Pap arrived with 300 mounted escorts, but quickly became worried by the absence of the emperor, who was still in Antioch, and therefore fled back toward Armenia and fought off a legion that was sent after him.

Terentius sent two generals with scutarii (shielded cavalry) familiar with the local terrain after Pap, an Armenian named Danielus and an Iberian named Barzimeres, who failed to capture Pap. The generals gave the excuse that Pap had used magical powers to avoid capture and used a dark cloud to mask his party, which is reminiscent of Faustus's claim that Pap was possessed by demons. This could have simply been an attack on Pap's character based on his sympathies towards Arians and pagans. Ammianus writes that Pap's subjects joyfully greeted their king's return, and that even after this assassination attempt Pap did not turn against the Roman Empire. Valens then ordered Traianus, Terentius's successor as comes et dux of Armenia, to gain Pap's confidence and murder him. A barbarian guard murdered Pap in 374 or 375 during a banquet which Traianus had organized for the young king. Ammianus describes the murder of Pap on Valens's orders as an unjustified and treacherous act, drawing parallels with the murder of the Quadi King Gabinius by Valentinian I and claiming that the ghost of Pap haunted many.

The Armenian still loyal to Pap did little to protest the murder of the king because of the large Roman army present in Armenian territory. The new Roman nominee for king, Pap's nephew Varazdat (Varasdates), was accepted by virtually everyone. Varazdat had grown up in Rome and began to rule under the regency of Mushegh Mamikonian, as the Mamikonians were the chief pro-Roman noble house in Armenia. Shapur II had long been courting Pap and the latter's murder and replacement with a Roman nominee provoked Persian outrage; however, Shapur did not invade and took only diplomatic action.

Marriage and issue
Pap married an Armenian noblewoman called Zarmandukht, who bore him two sons: Arshak (Arsaces) III and Vagharshak (Vologases). Pap's sons were later made co-rulers of Armenia by Manuel Mamikonian after he forced Pap's successor Varazdat to flee the country. Additionally, some scholars believe Varazdat to have been the illegitimate child of Pap.

Historiography
The classical Armenian historians are hostile to Pap and ascribe to him an array of sins, chief among which being the murder of Nerses I. Faustus of Byzantium, drawing from epical sources, depicts Pap as totally evil and possessed by demons from birth, which caused him to commit sins such as sodomy and bestiality. This attitude toward Pap has been explained by the king's troublesome relationship with the Armenian Church, caused by his promotion of Arianism and efforts to limit the church's power and influence. The Roman historian Ammianus Marcellinus, who, unlike the classical Armenian historians, was a contemporary of Pap, presents a more favorable image of the young king, whom he praises for his bravery and cleverness and describes as being welcomed “with the greatest joy by his subjects” after escaping the first assassination attempt against him. Later Armenian historians reevaluated Pap in a positive light, considering him an unjust victim of pro-church historians and valuing his attempts to strengthen the Armenian monarchy and pursue an independent foreign policy under difficult circumstances. Several of the "evil deeds" ascribed to Pap by Faustus were reinterpreted as the king's efforts to stabilize and repopulate the war-torn Kingdom of Armenia. Some other modern historians have evaluated Pap less positively. In Noel Lenski's view, Pap likely struggled to rule a kingdom that was still recovering from the destruction wrought by Shapur II, leading him to make poor decisions that ultimately led to his downfall.

Cultural depictions

 * Pap is a character in the tragedy Nerses the Great, Patron of Armenia written in 1857, by the 19th-century Armenian playwright, actor and editor, Sargis Vanandetsi (Sargis Mirzayan).
 * Pap is the titular character of the historical novel Pap Tagavor by Stepan Zoryan, first published in 1944.

title: Pap of Armenia


text:
ALCO RSC-3

The ALCO RSC-3 was a diesel-electric locomotive of the road switcher type rated at 1600 hp, that rode on three-axle trucks, having an A1A-A1A wheel arrangement.

Used in much the same manner as its four-axle counterpart, the ALCO RS-3, though the axle load was spread out for operation on light rail such as are found on branch lines. This locomotive had much better success as an export unit than in the domestic market.

Preserved
This list is incomplete. Please help expand it.

Australia
 * 4001 Preserved at the NSWRTM Thirlmere
 * 4002 Preserved at the PRHS Western Australia
 * 4006 On static display Wickham WA

Brazil
 * 7653 operating in Supervia

Canada
 * PGE 561 operating in Squamish at the West Coast Railway Heritage Park

Portugal
 * 1525 operating in SOMAFEL

title: ALCO RSC-3


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George Samuel Clason

George Samuel Clason (November 7, 1874 – April 5, 1957) was an  American author. He is most often associated with his book  The Richest Man in Babylon which was first published in 1926.

Early life and education
Clason was born in Louisiana, Missouri. He attended the University of Nebraska. He served in the United States Army during the Spanish–American War.

Career
Clason started two companies, the Clason Map Company of Denver, Colorado and the Clason Publishing Company. The Clason Map Company was the first to publish a road atlas of the United States and Canada, but did not survive the Great Depression.

Clason is best known for writing a series of informational pamphlets about being thrifty and how to achieve financial success. He started writing the pamphlets in 1926, using parables that were set in ancient Babylon. Banks and insurance companies began to distribute the parables, and the most famous ones were compiled into the book The Richest Man in Babylon - The Success Secrets of the Ancients. He is credited with coining the phrase, "Pay yourself first".

Personal life
Clason moved to Denver in 1900. He had two children who were also authors Clyde B. Clason and Robin Clason McKown. He was married twice, to Ida Ann Venable and Anna Burt.

He moved to Napa, California in 1949 and died there on April 5, 1957.

title: George Samuel Clason


text:
Murghab

Murghab (and various similar transliterations; Russian Мургаб) may refer to:
 * Marghab District, a district in Ghor province
 * Murghab River, rising from the Marghab district and ending in Turkmenistan
 * Bala Murghab District, a district in Badghis Province, northwestern Afghanistan
 * Murghab, Tajikistan, a town and district capital in Gorno-Badakhshan Autonomous Region, eastern Tajikistan
 * Murghob District, a district in Gorno-Badakhshan Autonomous Region, eastern Tajikistan
 * Murgap, Turkmenistan, a town in and district capital in Mary Province, Turkmenistan
 * Murgap District, a district in Mary Province, Turkmenistan
 * Bartang River, a river that rises in the Wakhan District of northeastern Afghanistan and flows into Tajikistan, which is known as "Murghab River" in its upper reaches

title: Murghab


text:
User talk:Aivar100~enwiki

Welcome!

Hello,, and welcome to Wikipedia. Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers: I hope you enjoy editing here and being a Wikipedian! By the way, please be sure to sign your name on talk pages using four tildes (~~~~) to produce your name and the current date, or three tildes (~~~) for just your name. If you have any questions, you can post to the help desk or ask me on my talk page. Again, welcome! --Jūzeris | Talk 16:47, 25 August 2005 (UTC)
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title: User talk:Aivar100~enwiki


text:
Bandhani

Bandhani is a type of tie-dye textile decorated by plucking the cloth with the fingernails into many tiny bindings that form a figurative design. Today, most Bandhani making centers are situated in Gujarat, Rajasthan, Sindh, Punjab region and in Tamil Nadu where it is known as Sungudi. It is known as chunri in Pakistan. Earliest evidence of Bandhani dates back to Indus Valley civilization where dyeing was done as early as 4000 B.C. The earliest example of the most pervasive type of Bandhani dots can be seen in the 6th century paintings depicting the life of Buddha found on the wall of Cave at Ajanta. Bandhani is also known as Bandhej Saree, Bandhni, Piliya, and Chungidi in Tamil and regional dialects. Other tying techniques include Mothra, Ekdali and Shikari depending on the manner in which the cloth is tied. The final products are known with various names including Khombi, Ghar Chola, Patori and Chandrokhani.

Etymology
The term bandhani is derived from the Sanskrit verbal root bandh ("to bind, to tie").

Overview

The art of Bandhani is a highly skilled process. The technique involves dyeing a fabric which is tied tightly with a thread at several points, thus producing a variety of patterns like Chandrakala, Bavan Baug, Shikari etcetera; depending on the manner in which the cloth is tied. The main colour used in Bandhana are yellow, red, blue, green and black. Each colour is traditionally tied to specific cultural meaning. Red is a symbol of marriage and is connected to rituals of married women, yellow stands for spring and is connected to both the season and childbirth, saffron is the colour of renouncer of the world and connects with warriors ready to give up their life in war or to yogis who give up worldly life, black and maroon are used for mourning.

As Bandhani is a tie and dye process, dying is done by hand and hence best colours and combinations are possible in Bandhanis. There are two types of dyeing traditionally categorised according to durability of colours - pakka, in which the colours do not come off easily and Kaccha, in which the colours fade or wash off easily. Historically, Kaccha technique was the more preferred one as the colours could be refreshed again and again while pukka technique was considered to be suitable for old people. The finest and most complicated patterns, whether for men's turbans or women's drapes called odhnis, were always dyed in kachcha colours. The main colours used in Bandhana are natural. T H Hendley, writing in the 19th century, provided the organic sources of the colours used for Bandhani, Most of them like the red ( both pukka and kaccha), indigo were derived from flowers while yellow by mixing turmeric with buttermilk.

In Gujarat, the Bandhani work has been exclusively carried out by the Khatri community of Kutchh and Saurashtra. A meter length of cloth can have thousands of tiny knots known as 'Bheendi' in the local language ('Gujarati'). These knots form a design once opened after dyeing in bright colours. Traditionally, the final products can be classified into 'khombhi', 'Ghar Chola', 'Chandrakhani', 'Shikari', 'Chowkidaar', 'Ambadaal' and other categories.

Bandhani work is also done in Rajasthan, where different colours and designs are used than the Kutch and Saurashtra regions of Gujarat. Establishments of varying sizes in the entire Kutch belt in Gujarat produce many varieties of Bandhani. This Bandhani style is called as the Kutchi Bandhani. The bold patterns of Bandhani are very similar in design, motifs, and technique in the desert belt encompassing northern Kutch in Gujarat, Western Rajasthan, and Sindh even in Pakistan.

Bandhani tying is often a family trade, and the women of these families work at home to tie patterns. Pethapur, Mandavi, Bhuj, Anjar, Jetpur, Jamnagar, Rajkot, are some of the main towns in Gujarat, where Bandhani is created. The city of Bhuj in Gujarat is well known for its red Bandhani. Dyeing process of Bandhani is carried out extensively in this city, as the water of this area is known to give a particular brightness to colours, specifically reds and maroons. As with other Indian textiles, in Bandhani too different colours convey different meanings. People believe that red is an auspicious colour for brides.

History
Earliest evidence of Bandhani dates back to Indus Valley civilization suggest that dyeing was done as early as 4000 B.C. The earliest example of the most pervasive type of Bandhani dots can be seen in the 6th century paintings depicting the life of Buddha found on the wall of Cave at Ajanta. This art finds its mentions in the Alexander the great time texts about the beautiful printed cottons of India. As per evidences in Historical Texts, the first Bandhani saree was worn at the time of Bana Bhatt's Harshacharita in a royal marriage. It was believed that wearing a Bandhani saree can bring good future to a bride. Ajanta walls stand for the evidences of these Bandhani sarees. The dyers have experimented with the use of different elements both natural and man-made for ages. Also, there are experiments with different binding/tying techniques to create patterns on cloth immersed in containers of dye. Different types of tie and dyes have been practiced in India.

Bandhej Saree

Bandhej saree which is also known as "Bandhani saree" is specially found in Gujarat and Rajasthan. As per the region of manufacturing the patterns of Bandhej saree may vary. Fine varieties of Bandhej are created in Pethapur, Mandvi, Bhuj, Anjar, Jamnagar, Jetpur, Porbandar, Rajkot, Udaipur, Jaipur, Ajmer, Bikaner, Churu etc. They are considere prized possessions of married women and are mostly essential part of traditional bridal trousseau. In Rajasthan and Gujarat, Bandhani fabrics are very popular with men and women but bandhani saree from a ritual necessity for married women for many ceremonies. Many Gujarati brides wear Gharchola, a type of Bandhej saree, for their weddings. Though Gharchola literally means "robe for home", in ritual parlance, it means "attire for the new home or the home of the husband" and is usually a gift to the bride form her mother-in-law. In Rajasthan, during pregnancy or childbirth, the natal home gifts the women peeley ki saree. It is a combination of yellow base with a broad red border with bandhani pattern on it.

Bandhani, one of the oldest known methods of tie-dyeing, is still widely practiced in western India today. The fabric is made by pinching very small portions of cloth and tying them by plucking the cloth with the fingernails into many tiny bindings that form a figurative design to form an intricate pattern of dots. The cloth is then placed into different dye vats to form bright and beautiful colors.

Process
Bandhani is a method of binding small knots and dyeing them in different colours to produce beautiful patterns. This tying was normally done with fingernails for making. But in some places of Rajasthan, craftsmen wear a metal ring with a pointy nail to help plucking the cloth easily.

The process of making a bandhani textile is not very difficult, but is very time-consuming. The fabric used for making Bandhani sarees and dupattas are loosely woven silk called Georgette, or cotton known as Malmal. The knots are tightly tied, and the rest of the fabric is dyed in multiple stages. This leaves the knots undyed and hence a beautiful flower-like pattern appears all over the cloth as a design.

Mulmul (fine muslin), handloom or silk cloth were the traditional choices but now chiffon, georgette and crepe are also being used as base fabrics for Bandhani. This cloth is washed to remove traces of starch, and then bleached to attain a clear base. It is then folded into two or four layers depending on the thickness of the cloth. A designer marks the layout of the pattern on the material using wooden blocks dipped in geru, a natural clay earth pigment mixed with water. The cloth is tied from the areas not to be dyed. The process requires patience, expertise and meticulousness on part of the artist. The folds of the material within the small motif have to be lifted and tied together. The material with the first set of ties is dyed yellow. The material is again tied and dyed into red or green. The artist moves from lighter to darker shades and the use of more and varied colours makes the process complicated. If the border has to be darker all the lighter parts are tied and covered with plastic foil and the edges are dyed with the required colors. Repeated tying and dyeing produces elaborate designs. The designs may run into a single motif and or a combination of large and small motif alternating in some order.

title: Bandhani


text:
User:Hanro~enwiki

GLOBAL WARMING AND ITS EFFECTS!

In the year 2025 due to global warming. Many parts of the united kingdom will unfortunately be flooded. However it is not bad as many lesser towns will disappear alleviating the country of many of its problems. Just think with slough and hounslow and also feltham buried under water we can expect a fall in crime rates of approximately 32.5%. Another point worth noting is that many areas of london which could be described as eyesores blotting this wonderful country will also go. This is fact we have this from several leading academic institues including the university of slough itself. Please note howeve that this is only one of many rumored possibilities for the future. Obviously it might not happen but we should be hopeful.

In addition to the previous towns i happy to say would disappear we must rejoice at the prospect of losing my favourite town HULL!!! Hanro Patango and ................ the infamous........ Hanro Patango's assistant

title: User:Hanro~enwiki


text: User talk:70.174.32.75

title: User talk:70.174.32.75


text:
User:Paul D. Anderson/sandbox

A uniform polyhedron is a polyhedron with regular polygon faces and identical vertices. Uniform polyhedra are semi-regular but the faces and vertices need not be convex.

Exclusive of the infinite sets of prisms and antiprisms, there are 75 uniform polyhedra.

The convex uniform polyhedra include the five Platonic solids, the thirteen Archimedean solids, and the convex prisms and antiprisms.

The non-convex uniform polyhedra include the four Kepler-Poinsot solids, 53 additional non-convex uniform polyhedra, and the non-convex prisms, antiprisms and crossed antiprisms.

title: User:Paul D. Anderson/sandbox


text:
WBAB

WBAB (102.3 FM) is a classic rock radio station licensed to Babylon, New York and owned by Cox Radio. The station is also simulcast on WHFM (95.3 FM) licensed to Southampton, New York and serving eastern Long Island.

History
WBAB first went on the air August 27, 1958, as WBAB-FM. It simulcast WBAB (1440 AM), until September 1975 when 1440 AM adopted a Gospel music format.

2006 signal hijacking
On the morning of Wednesday, May 17, 2006, the station's signal was hijacked for about 90 seconds while the signal jammers broadcast the song "Nigger Hatin' Me" by the 1960s-era white supremacist country music singer Johnny Rebel. Roger Luce, the station's morning host, said at the time, "I've never seen this in 22 years at this radio station... Whatever that was - it was very racist."

The next morning, it made the front page of Newsday with the headline "JACKED FM". The station's new general manager, John Shea, said, "I've only been here a week and we get hijacked." Former program director John Olsen said, "This was not some child's prank, this was a federal offense."

The hijack was likely accomplished by overpowering the studio transmitter link (STL) signal to the transmitter in Dix Hills, New York. A signal hijacking with the same song happened to WBAB's sister station WBLI about two weeks earlier.

Discography

 * WBAB Homegrown Album (1981)
 * WBAB Son of Homegrown (1984)

title: WBAB