Bernard Dussardier
email: bernard.dussardier@unice.fr

Wilfried Blanc

Gérard Monnom

Luminescent ions in silica-based optical fibers

Keywords: optical fiber, silica, spectroscopy, rare-earths, transition metals, energy transfer, valence state, phonon energy, local environment

We present some of our research activities dedicated to doped silica-based optical fibers, aiming at understanding the spectral properties of luminescent ions, such as rare-earth and transition metal elements. The influence of the local environment on dopants is extensively studied: energy transfer mechanisms between rare-earth ions, control of the valence state of Chromium ions, effect of the local phonon energy on Thulium ions emission efficiency and broadening of Erbium ions emission induced by oxide nanoparticles. Knowledge of these effects is essential for photonics applications.

Introduction

During the last two decades, the development of sophisticated optical systems and devices based on fiber optics have benefited from the development of very performant optical fiber components. In particular, optical fibers doped with 'active' elements such as rare-earth (RE) ions have allowed the extremely fast development of optical telecommunications [i,ii], lasers [iii] industries and the development of temperature sensors [iv]. The most frequently used RE ions (Nd 3+ , Er 3+ , Yb 3+ , Tm 3+ ) have applications in three main spectral windows: around 1, 1.5 and 2µm in fiber lasers and sensors based on absorption/fluorescence and around 1.5 µm for telecommunications and temperature sensors. RE-doped fibers are either doped with one element (e.g. Er 3+ in line amplifiers for long haul telecommunications) or two elements (e.g.

Yb 3+ and Er 3+ in booster amplifiers or powerful 1.5 µm lasers). In the second case, the nonradiative energy transfer mechanism from donor to acceptor is implemented to benefit from the good pump absorption capacity of the donor (e.g. Yb 3+ around 0.98 µm) and from the good stimulated emission efficiency of the acceptor (e.g. Er 3+ around 1.5 µm). All the developed applications of amplifying optical fibers are the result of long and careful optimization of the material properties, particularly in terms of dopant incorporation in the glass matrix, transparency and quantum efficiency.

The exploited RE-doped fibers are made of a choice of glasses: silica is the most widely used, sometimes as the result of some compromises. Alternative glasses, including low Maximum Phonon Energy (MPE) ones, are also used because they provide better quantum efficiency or emission bandwidth to some RE ions particular optical transition. The icon example is the Tm 3+ -Doped Fiber Amplifier (TDFA) for telecommunications in the S-band (1.48-1.53 µm) [v], for which low MPE glasses have been developed: oxides [vi,vii], fluorides [viii], chalcogenides [ix]… However, these glasses have some drawbacks not acceptable at a commercial point of view: high fabrication costs, low reliability, difficult connection to silica components and, in the case of fiber lasers, low optical damage threshold and resistance to heat.

To our knowledge, silica glass is the only material able to meet most of applications requirements, and therefore the choice of vitreous silica for the active fiber material is of critical importance. However a pure silica TDFA would suffer strong non-radiative de-excitation (NRD) caused by multiphonon coupling from Tm 3+ ions to the matrix. Successful insulation of Tm 3+ -ions from matrix vibrations by appropriate ion-site 'engineering' would allow the development of a practical silica-based TDFA.

Other dopants have recently been proposed to explore amplification over new wavelength ranges. Bi-doped glasses with optical gain [x] and fiber lasers operating around 1100-1200 nm have been developed [xi,xii], although the identification of the emitting center is still not clear, and optimization of the efficiency is not yet achieved. Transition metal (TM) ions of the Ti-Cu series would also have interesting applications as broad band amplifiers, super-fluorescent or tunable laser sources, because they have in principle ten-fold spectrally larger and stronger emission cross-sections than RE ions. However, important NRD strongly reduces the emission quantum efficiency in silica. Bi-and TM-doped fibers optical properties are extremely sensitive to the glass composition and/or structure to a very local scale. As for Tm 3+ ions, practical applications based on silica would be possible when the 'ion site engineering' will be performed in a systematic approach. This approach is proposed via 'encapsulation' of dopants inside glassy or crystalline nanoparticles (NP) embedded in the fiber glass, like reported for oxyfluoride fibers [xiii] and multicomponent silicate fibers [xiv]. In NP-doped-silica fibers, silica would act as support giving optical and mechanical properties to the fiber, whereas the dopant spectroscopic properties would be controlled by the NP nature. The NP density, mean diameter and diameter distribution must be optimized for transparency [xv].

In this context, our group has made contributions in various aspects introduced above.

Our motivations are both fundamental and application oriented. First, the selected dopants act as probes of the local matrix environment, via their spectroscopic variations versus ligand field intensity, site structure, phonon energy, statistical proximity to other dopants,… The studies are always dedicated to problems or limitation in applications, such as for Erbium-Doped Fiber Amplifier (EDFA) and TDFA, or high temperature sensors. It is also important to use a commercially derived fabrication technique, here the Modified Chemical Vapor Deposition (MCVD), to assess the potential of active fiber components for further development.

The aim of this paper is reviewing our contributions to improving the spectroscopic properties of some RE and TM ions doped into silica. The article is organized as follows:

Section 0 describes the MCVD fabrication method of preform and fiber samples, and the common characterization techniques used in all studies. Section 0 Chapitre: is devoted to the study of energy transfers in Erbium ion (Er 3+ ) and Ytterbium:Erbium (Yb 3+ :Er 3+ ) heavily (co)doped fibers and the applications to fiber temperature sensors, whereas Section 0 summarizes our original investigations on Chromium (Cr 3+ and Cr 4+ ) in silica-based fibers. In Section 0 Chapitre:, we report on the spectroscopic investigations of Thulium-(Tm 3+ ) doped fibers versus the material composition, including phonon interactions and non-radiative relaxations. In section 0 Chapitre: are reported our recent discoveries in RE-doped dielectric nanoparticles, grown by phase separation.

Experimental Preforms and fibers fabrication

All the fibers investigated in this article were drawn from preforms prepared by the Modified Chemical Vapor Deposition (MCVD) technique [xvi] at Laboratoire de Physique de la Matière Condensée (Nice). In this process, chemicals (such as O 2 , SiCl 4 ) are mixed inside a glass tube that is rotating on a lathe. Due to the flame of a burner moving along the tube, they react and extremely fine silica particles are deposited on the inner side of the tube. These soot are transformed into a glass layer (thickness is about few µm) when the burner is passing over.

The cladding layers are deposited inside the substrate tube, followed by the core layers.

Germanium and Phosphorus can be incorporated directly through the MCVD process. They are added to raise the refractive index. Moreover, this last element is also added as a melting agent, decreasing the melting temperature of the glass. All the other elements (rare-earths, transition metals, aluminium, …) are incorporated through the solution doping technique [xvii]. The last core layer is deposited at lower temperature than the preceding cladding layers, so that they are not fully sintered and left porous. Then the substrate tube is filled with an alcoholic solution of salts and allowed to impregnate the porous layers. After 1-2 hours, the solution is removed, the porous layer is dried and sintered. When the deposition is complete, the tube is collapsed at 2000°C into a preform. In our case, the typical length of the preform is about 30 cm and the diameter is 10 mm. The preform is then put into a furnace for drawing into fiber. The preform tip is heated to about 2000°C. As the glass softens, a thin drop falls by gravity and pulls a thin glass fiber. The diameter of the fiber is adjusted by varying the capstan speed. A UV-curable polymer is used to coat the fiber.

Material characterizations

Refractive index profiles (RIP) of the preforms were measured using a York Technology refractive index profiler (P101), while the RIPs of the optical fibers were determined using a York Technology refractive index profiler (S14). The oxide core compositions of the samples were deduced from measurement of the RIP in the preform, knowing the correspondence between index rising and AlO 3/2 , GeO 2 , PO 5/2 concentration in silica glass from the literature [xviii,xix]. The composition was also directly measured on some preforms using electron probe microanalysis technique in order to compare results. A good agreement was found. The concentration of these elements is generally around few mol%. Luminescent ions concentrations are too low to be measured through the RIP. They were measured through absorption spectra.

For example, Tm 3+ ion concentration has been deduced from the 785 nm ( 3 H 6 => 3 H 4 ) absorption peak measured in fibers and using absorption cross-section reported in [xx]:  abs (785 nm) = 8.7x10 -25 m 2 .

Energy transfers in Er 3+ and Yb 3+ :Er 3+ heavily doped silica fibers

The non-radiative energy transfer processes are well-known phenomena that influence the optical properties of doped-materials. The first theoretical basis appears in the 50's with the Förster-Dexter's model [xxi,xxii] that treats this process as the result of dipole-dipole and multipole-multipole interactions. Two energy transfer processes are described in Fig. 1. When pumping a co-doped material some ions are promoted in one of their excited level. If some ions are close to each other, their wave-functions interpenetrate and the energy stored in the excited level of the donor ions is non-radiatively transferred to a resonant level of the acceptor ions. This process was turned to good account in Yb 3+ :Er 3+ co-doped silica fibers for high power fiber amplifier [xxiii] and laser [xxiv] applications : it takes advantage of the strong absorption crosssection of Yb 3+ at 980 nm and of the high efficiency of the energy transfer. In the case of high doping levels for both species, another non-radiative energy transfer process can take place and allows exciting a higher level of the acceptor ion : it is the double energy transfer process (DET) described by Auzel [xxv]. This process was first used to convert infrared light from LED to visible emission or to detect weak infrared signals with photomultipliers [xxvi,xxvii].

Double energy transfer in Er 3+ -doped fibers

The clustering effect in Er 3+ -doped silica fibers is now a well-accepted phenomenon, and its detrimental influence on the 1550-nm gain transition of such fibers is well established [xxviii]. For simplicity, modeling of clusters has consisted of considering that a fraction of the dopants were organized in ion pairs [xxix], in which an immediate energy transfer leads to an instantaneous relaxation of one excited ion. This model is in very good agreement with the experimental results obtained for saturable absorption and for gain measurements at low Er 3+ doping levels as in fiber amplifiers. At higher doping levels, Ainslie et al. [xxx] showed that, in addition to the ions dispersed in the host, regions in which concentrations of rare-earth exceeding 40 wt% -called clusters -appear : in such a material the ion-pair model cannot be applied. We have developed a cluster model [xxxi] that differs from the ion-pair one by the fact that we consider that each ion of a cluster can efficiently transfer its energy to any of the other ions of the same cluster. When n ions of a cluster are excited, a succession of (n-1) fast relaxations by energy transfer leads to a situation in which all the ions of a cluster but one are de-excited. This model permits the determination of the proportion of the dopants organized in clusters and the transfer rate. In order to validate the model we realized a pump-absorptionversus-pump-power experiment with two fiber samples, Er-1 and Er-2, doped with 100 ppm and 2,500 ppm of Er 3+ , respectively (Fig. 2). This shows that the non-saturable absorption (NSA) grows dramatically with the Er 3+ concentration. We have attributed this behaviour to the presence of clusters containing a significant percentage of the dopants and in which efficient energy transfers allow these ions to relax rapidly after the absorption of a first pump photon.

Double energy transfer in highly Yb 3+ :Er 3+ -co-doped fibers

The green fluorescence of Er 3+ -doped optical fiber is a well-known phenomenon in 800 nm-pumped erbium-doped fibers. This emission results from the excited state absorption phenomenon and is characteristic of the emission from the 2 H 11/2 and 4 S 3/2 -levels and consequently can be observed with any pumping scheme leading to the population of these levels. We have studied how these levels can be excited by DET [xxxii] and a schematic energy diagram is shown in Fig. 3.

At low rare-earth concentrations, the large inter-ionic distances permit efficient single energy transfers, but the second energy transfer is very inefficient. For applications in which the green fluorescence is desirable, this second energy transfer must be enhanced. For that the rareearth concentration must be as high as possible to reduce the distance between neighboring ions.

In this case, a second phase, referred to as clusters, can appear in which the rare-earth ions concentration is particularly high. In order to quantify the fraction of active ions into clusters, we have studied the Yb 3+ and Er 3+ fluorescence dynamics in a highly co-doped fiber ([Yb]=[Er]=2,500 ppm) : the 1040 nm-fluorescence decay represents the population decay from the 2 F 5/2 -metastable level of Yb 3+ , and that of the green-fluorescence represents the evolution of the 2 H 11/2 and 4 S 3/2 -populations of Er 3+ . Our setup allows simultaneous measurements of the counter-propagative visible emission and the lateral infrared emission. The experimental curves show two typical decays. Fitted with our rate equations model [xxxii], they revealed that roughly 50% of both ions are organized in clusters in the co-doped fiber. This high percentage must be associated with very high Yb-Er transfer rates (3x10 6 s -1 ), one order of magnitude superior to the Er 3+ : 4 I 11/2 intermediate level relaxation rate (3.7x10 5 s -1 ): Er 3+ ions placed in their short lived 4 I 11/2 state have a higher probability to be excited to the 4 F 7/2 upper state than to relax spontaneously. The strong percentage of ions organized in clusters and the very high transfer rates are at the origin of the very good up-conversion efficiency.

Thermalization effects between excited levels in doped fibers: temperature sensor based on fluorescence of Er 3+

Though the rare-earth ions are never in thermodynamical equilibrium because of the metastability of some levels, it has been demonstrated that the populations of the 2 H 11/2 and 4 S 3/2 levels responsible for the green emission in Er 3+ -doped fibers are in quasi-thermal equilibrium. This effect has been observed for the first time in fluoride glass fibers [xxxiii] and can be attributed to the relatively long lifetime of these levels (400 µs) in that host. In silica, in spite of the two orders of magnitude shorter lifetime, a fast thermal coupling between both levels has been proposed [xxxiv] and confirmed experimentally [xxxv] (Fig. 4). Indeed these levels can be considered to be in quasi-thermal equilibrium, because of the small energy gap between them, about 800 cm -1 , compared to the high energy gap between them and the nearest lower level, about 3000 cm -1 . In this case, the lifetime of these levels is sufficient (1 µs) to allow populating the upper level from the lower one by phonon induced transitions. Therefore R, the ratio of the intensities coming from both levels, can be written as:

R  I 2 H 11/2   I 4 S 3/ 2     2 H 11/ 2    4 S 3/2    e 2 H 11/2    e 4 S 3/2   exp E / kT   (1)
where  is the frequency,  e the emission cross section, k the Boltzmann constant, E the energy gap between the two levels and T the temperature in degrees Kelvin. In Fig. 4 we show that the experimental data can be fitted by a function in agreement with Equation (1). This is another example of an energy transfer process, this one being assisted by phonons.

In order to take advantage of the high efficiency of the DET in highly co-doped Yb-Er doped fiber and of the thermalisation effect between the higher levels involved in the green fluorescence in this kind of fiber, we have developed a new temperature sensor, unsensitive for strain. The dynamic obtained was 11 dB in Fig. 4 over the shown temperature range, leading to a mean rate of change of the green intensity ratio of approximately 0.016 dB/K at 300 K.

Several temperature cycles have been carried out and we have observed a good repeatability. As for the stability, no modifications have been observed on the two intensities when the fiber was heated during several hours at temperatures up to 600°C. Due to the strong absorption of the doped fiber in the signal wavelength range -the green emissions corresponding to transitions downto the fundamental level -and to the 15 dB/km intrinsic absorption of the transparent fiber in the same wavelength range, such a device would be limited to a point sensor.

We have developed a new sensor based on the 1.13 µm and 1.24 µm emission lines, coming from the same levels [xxxvi]. These lines present the same temperature behaviour as the green ones. As the lower level of these transitions is the 4 I 11/2 -level and not the fundamental one (Fig. 3), the signals are absorption free and their wavelengths correspond to a transparency region of the intermediate fibers. These arguments have permitted the development of an efficient quasi-distributed configuration without limitation on the sensing line length : the short lifetime of upper levels (1 µs) could allow realizing a sensors network. Each sensitive head is separated from its neighbors by a 100-meter long transparent silica fiber in order to time-resolve the counter-propagative signals.

Conclusion

Energy transfer processes in rare-earth-doped materials have been studied since the middle of the 20 th century. At the beginning, the applications of DET were mainly conversion of infrared light to visible emission or detection of weak infrared signals with photomultipliers. A renewal interest appears with the development of optical fibers in which high power density can be achieved: single energy transfer allows improvement of high power fiber amplifiers and laser and DET permits realizing point and quasi-distributed fiber sensors.

Local structure, valency states and spectroscopy of transition metal ions

Optical fiber materials with very broad-band gain are of great interest for many applications. Some reports on chromium-doped glasses have already shown evidence of absorption and nearinfrared (NIR) fluorescence due to Cr 4+ in these materials [xl,xli]. However their compositions and preparation techniques greatly differ from those of silica optical fibers. Therefore, some basic studies on the optical properties of TM ions in silica-based optical fibers are needed. In particular, the final TM oxidation state(s) in the fiber core strongly depend(s) on the preparation process. Also, the optical properties (absorption and luminescence) of one particular oxidation state of a TM ion varies from one host composition and structure to another, due to variations of the crystal-field (so-called ligand field in glass) [xlii]. Hence the interpretation of absorption and emission spectroscopy is difficult. Because no luminescence spectroscopy of the TM-doped silica fibers had been reported before, we have contributed to explore this field. We have studied the influence of the chemical composition of the doped region on the Cr-oxidation states and on the spectroscopic properties of the samples. We have also studied the optical properties versus the experimental conditions (temperature and pump wavelength). We describe the experimental details specifically used for TM-doped fibers, then we summarize all results and interpretations.

Fabrication and characterization of Chromium-doped samples

The preforms and fibers were prepared as described in §0 Chapitre:, using Cr 3+ -salt alcoholic doping solution and oxygen or nitrogen (neutral) atmosphere for the drying-tocollapse stages. Three different types of samples containing Ge or/and Al were prepared, referred to as Cr(Ge), Cr(Ge-Al) and Cr(Al), respectively. The total chromium concentration ([Cr]) was varied from below 50 mol-ppm to several thousands mol ppm. Above several 100 s mol ppm, preform samples had evidence of phase separation causing high background optical losses, whereas fibers (few 10s mol ppm) did not show phase separation and had low background losses (<1 dB/m). The oxidation states of Cr and their relative concentrations were analyzed by Electron Paramagnetic Resonance, whereas the absolute content of all elements (including Cr) was analyzed by Plasma Emission Spectroscopy.

Absorption spectra were analyzed using the Tanabe-Sugano (T.-S.) formalism [xliii] to compare our assignments to optical transitions with reports on Cr 3+ -and Cr 4+ -doped materials.

This formalism helps predicting the energy of electronic states of a TM in a known ligand field symmetry as a function of the field strength Dq and the phenomenologic B and C so-called

Racah parameters (all in cm -1 , Fig. 5). The Dq/B ratio allows the qualitative determination of some optical properties of TM ions, such as strength, energy and bandwidth of optical transitions. We have also estimated the absorption cross-sections using results from composition and valency measurements. Absorption and emission spectroscopies including decay measurements were performed on both preforms and fibers, at room temperature (RT) and low temperature (LT, either 12 or 77 K), using various pump wavelengths: 673, 900 and 1064 nm.

Full details of the experimental procedures are given in [xliv,xlv,xlvi].

Principal results

By slightly modifying the concentration in germanium and/or aluminium in the core of the samples, their optical properties are greatly modified. In particular, we have shown that: ii) The absorption spectra have been interpreted and optical transitions assigned for each present valency state (Fig. 6). The absorption cross sections curves ( abs ) were estimated. For Cr 3+ ,  abs (Cr 3+ , 670 nm)= 43 x 10 iii) Using the T.-S. formalism, we found Dq/B = 1.43 which is lower than the value were 3 T 2 and 1 E levels cross (Dq/B = 1.6, Fig. 5). As a consequence, the expected emission is along the 3 T 2  3 A 2 transition as a broad featureless NIR band. No narrow emission line from the 1 E state is expected, in agreement with fluorescence measurements. Dq/B is lower than those reported for Cr 4+ in laser materials like YAG and Forsterite [xlviii].

iv) The LT fluorescence from Cr 4+ spreads over a broad spectral domain, from 850 to 1700 nm, and strongly varies depending on core chemical composition, [Cr] and  p (pump wavelength). The observed bands were all attributed to Cr 4+ ions, in various sites. Fig. 7 shows the fluorescence spectra of Cr 4+ in two different types of samples and in various experimental conditions. Possible emission from other centers (Cr 3+ , Cr 5+ , Cr 6+ ) was discussed, but rejected vi) The LT fluorescence decays are non-exponential (Fig. 8) and depend on [Cr] and  s .

The fast decay part is assigned to Cr clusters or Cr 4+ -rich phases within the glass. The 1/elifetimes ( at  s = 1100 nm are all within the15-35 s range in Al-containing samples, whereas  ~ 3-11 s in Cr(Ge) samples, depending on [Cr]. The lifetime of isolated ions ( iso ), measured on the exponential tail decay curves (not shown) reach high values:  iso ~200 to 300 µs at  s ~1100 nm,  iso ~70 µs at  s ~1400 nm. In the heavily-doped Cr(Ge) samples,  iso is an order of magnitude less. Hence, Cr 4+ ions are hosted in various sites: the lowest energy ones suffer more non-radiative relaxations than the higher energy ones. Also presence of Al improves the lifetime, even at high [Cr]. It is estimated that at RT, lifetime  would be of the order of 1 µs or less. This fast relaxation time, compared to RE ions (~1 ms) has been implemented as a fiberized saturable absorber in a passively Q-switched all-fiber laser [liv].

Conclusion

The

Oxide modifiers influence on the 3 H 4 -level lifetime

To address this problem, we have studied the effect of some modifications of Tm 3+ ion local environment. Keeping the overall fiber composition as close as possible to that of a standard silica fiber, we expect to control the rare-earth spectroscopic properties by co-doping with selected modifying oxides. We have studied the incorporation of modifying elements compatible with MCVD. GeO2 and AlO 3/2 are standard refractive index raisers in silica. AlO 3/2 is also known to improve some spectroscopic properties of Er 3+ ion for C-band amplification [i] and to reduce quenching effect through clustering in highly rare-earth-doped silica [lvii]. Both oxides have a lower maximum phonon energy than silica. We use high phonon energy PO 5/2 as opposite demonstration. GeO2 and PO 5/2 concentrations are 20 and 8 mol%, respectively. AlO 3/2 concentration is varied from 5.6 to 17.4 mol%. Tm 3+ concentration is less than 200 mol ppm.

To investigate the role of the modification of the local environment, decay curves of the 810 nm fluorescence from the 3 H 4 level were recorded. All decay curves measured are nonexponential. This can be attributed to several phenomena and will be discussed in this article.

Here, we study the variations of 1/e lifetimes () versus concentration of oxides of network modifiers (Al or P) and formers (Ge). The lifetime strongly changes with the composition of the glass host. The most striking results are observed within the Tm(Al) sample series:  linearly increases with increasing AlO 3/2 content, from 14 µs in pure silica to 50 µs in sample Tm(Al) containing 17.4 mol% of AlO 3/2 . The lifetime was increased about 3.6 times. The lifetime of the 20 mol% GeO 2 doped fiber Tm(Ge) was increased up to 28 µs whereas that of the 8 mol% PO 5/2 doped fiber Tm(P) was reduced down to 9 µs. We see that aluminum codoping seems the most interesting route among the three tested codopants.

Non-exponential shape of the 810-nm emission decay curves

All fluorescence decay curves from the 3 H 4 level are non-exponential. We have investigated the reasons for this non-exponential shape in various silica glass compositions. We observed that the decay curve shape depends only on the Al-concentration, even in the presence

I(t)  A i exp t /  i   i  (2)
where A() is the continuous distribution of decay constant.

The procedure for calculating  and the fitting algorithm are described in detail in [lix]. For the fitting procedure, we considered 125 different values for  i , logarithmically spaced from 1 to 1000 µs. By applying this procedure to all the decay curves, a good matching was generally obtained. For a given composition (Fig. 10), we can notice two main distributions of the decay constant. With the aluminium concentration, they increase from 6 to 15 µs and from 20 to 50 µs, respectively. For the highest aluminium concentration (9 mol%, in Tm(Ge) and Tm(P)), these two bands are still present (not shown in the figure). One is around 10 µs and the second one spreads from 30 to 100 µs, for both compositions (Tm(Ge) and Tm(P)). According to the phenomenological model, the width of the decay constants distribution is related to the number of different sites. The large distribution around 80 µs is then due to a large number of sites available with different environments. It is however remarkable that this distribution at 80

µs is very similar in both sample types. From the Tm 3+ ion point of view (considering luminescence kinetics), Tm(Ge) and Tm(P) glasses seem to offer the same sites.

The meaning of the decay constant values is now discussed. Lifetime constants obtained from the fitting can be correlated with the one expected for Thulium located in a pure silica or pure Al 2 O 3 environment. The 3 H 4 lifetime is calculating by using this equation:

1   1  rad W nr (3)
where  rad corresponds to the radiative lifetime which is given to be 670 µs in silica [lx].

W nr is the non-radiative decay rate, expressed as [lxi]:

W nr  W 0  exp (E  2E p )   (4)
where W 0 and  are constants depending on the material, E is the energy difference between the 3 H 4 and 3 H 5 levels and E p is the phonon energy of the glass. W 0 and  were estimated for different oxide glasses [lxi, lxii]. The energy difference E was estimated by measuring the absorption spectrum of the fibers. When Al concentration varies, this value is

almost constant around 3700 cm -1 [lxiii].
With these considerations, the 3 H 4 expected lifetime can be calculated. In the case of silica glass,  silica = 6 µs and for an Al 2 O 3 environment,  alumina = 110 µs. These two values are in accordance with the ones we obtained from the fitting procedure. The distribution of decay constant around 10 µs corresponds to Tm 3+ ions located in almost pure silica environment while the second distribution is attributed to Tm 3+ located in Al 2 O 3 -rich sites.

Conclusion

By adding oxide network modifiers or formers, we demonstrated that aluminium is the most efficient to improve the 3 H 4 level-lifetime. This was attributed to a lower local phonon energy. Potential of the amplification in the S-band was then investigated. In the fiber with the highest aluminium concentration, gain curve was measured. Although excitation wavelength (1060 nm), refractive index profile and Thulium concentration were not optimized, a gain of 0.9

dB was obtained at 1500 nm [lxiv]. With a numerical model of the TDFA that we developed

[lxv], we estimated that a gain higher than 20 dB is reachable in a silica-based TDFA.

Rare-earth-doped dielectric nanoparticles

Erbium-doped materials are of great interest in optical telecommunications due to the Er 3+ intra-4f emission at 1.54 µm. Erbium-Doped Fiber Amplifiers (EDFA) were developed in silica glass because of the low losses at this wavelength and the reliability of this glass. Developments of new rare-earth doped fiber amplifiers aim to control their spectroscopic properties: shape and width of the gain curve, optical quantum efficiency, .... Standard silica glass modifiers, such as aluminum, give very good properties to available EDFA. However, for more drastic spectroscopic changes, more important modifications of the rare-earth ions local environment are required. To this aim, we present a fiber fabrication route creating rare-earth doped calcosilicate or calco-phospho-silicate nanoparticles (NP) embedded in silica glass.

Nanostructured fibers preparation

In the chosen route, NP are not prepared ex-situ and incorporated into the perform. To prepare them, we take advantage of the heat treatement occurring during the MCVD process.

Their formation is based on the basic principle of phase separation. On the basis of thermodynamical data such as activity coefficient, entropy of mixing, enthalpy of mixing and Gibbs-free energy change, the phase diagram of the SiO 2 -CaO binary compound was derived using Factstage software (Fig. 11). A miscibility gap is found when the CaO concentration is between 2 and 30 mol%. In this region, CaO droplets are formed, like oil in water. Such phenomenon is expected during perform fabrication as temperature reaches 2000°C during collapsing passes.

For Calcium doping, CaCl 2 salt was added to the Er 3+ containing soaking solution. Four CaCl 2 concentrations were studied (0, 0.001, 0.1 and 1 mol/l). Ge and P were also added by MCVD. When the Ca concentration was increased in the doping solution, the aspect of the central core of the preform turned from transparent to milky. This variation is explained by the structural changes of the core. For preforms with calcium concentration higher than 0.01 mol/l, NP were observed by Transmission Electron Microscopy (TEM) on preform samples (Fig. 12).

We can clearly observe polydisperse spherical NP with an estimated mean diameter of 50 nm.

Smaller particles of 10 nm are visible. The size of the biggest particles was around 200 nm (not shown in Fig. 12). When the Ca concentration decreases, the size distribution of the particles is nearly identical but the density is lower. The composition of the core was investigated by Energy Dispersive X-ray analysis: the NP contained equal amounts of Ca, P and Si cations, whereas only Si cations was detected in the surrounding matrix. Ge seemed to be homogeneously distributed over the entire glass. The most important finding is that Er 3+ ions and Ca were detected only within the NP.

Erbium emission characterizations

Spectroscopic characterizations on the emission line associated to the 4 I 13/2 -4 I 15/2 transition at 1.54 µm were made at room temperature on Er-doped samples with (sample A) and without (sample B) calcium. The results are shown in Fig. 13 where we evidence the fact that the emission spectrum of sample A is broader than that of sample B. To explain these differences we have studied the Er 3+ local environment. EXAFS measurements at the Er-L III edge (E=8358 eV) were carried out at the GILDA-CRG beamline at the European Synchrotron Radiation Facility. In sample A the rare-earth is linked to O atoms in the first coordination shell and to Si or P atoms (these two atoms can not be distinguished due to the similar backscattering amplitude and phase) in the second shell in a way similar to that already observed in silicate glasses [lxvi], phosphate glasses [lxvii]. Also the structural parameters (about 7 O atoms at 2.26 Å and Si (or P) atoms at 3.6 Å corresponding to a Er-O-Si (or P) bond angle of ≈140 deg) are in good agreement with the cited literature. Si(or P) atoms are visible as they belong to the same SiO 4 (or PO 4 ) tetrahedron as the first shell O atoms but no further coordination shells are detected. This permits to state that an amorphous environment is realized around Er 3+ ions. On the other hand sample B presents a completely different EXAFS signal that is well comparable with the spectrum of ErPO 4 . This means that Er in this case is inserted in a locally well ordered phase of about a few coordination shells (around 4-5 Å around the absorber). The fact that TEM on this sample reveals a uniform sample is not in contradiction with this result; it just means that this phase is not spatially extended to form nm-sized NP (in our TEM analyses the spatial resolution is limited to few nm) but the ordering is extremely local, i.e. it is limited to only a few shells around the rare-earth ion.

From these considerations, the broadening of the emission spectrum observed in sample A can be attributed to an inhomogeneous broadening due to Er 3+ ions located in a more disordered environment compare to sample B. Here we see that the cumulated effects of Ca and P within the Er-doped NP both amorphize the material structure around Er 3+ ions and increase the fluorescence inhomogeneous broadening.

Conclusion

In this paragraph, we have demonstrated that through the phase separation mechanism, nanoparticles can be obtained in preforms by adding Calcium. Er 3+ ions are found to be located only into these nanoparticles. An inhomogeneous broadening of the emission band is observed, associated to Er 3+ ions located in a more disordered environment compare to silica. This feature is particularly interesting in the production of materials for Wavelength Division Multiplexing applications, such as Erbium-Doped Fiber Amplifier with a broader band gain.

Perspectives and conclusion

The choice of a glass to develop new optical fiber component is most of the time a result of compromises. Silica glass is the most widely used for its many advantages (reliability, low cost fabrication, …). However, it suffers from different drawbacks, such as high phonon energy or low luminescent ions solubility, which affect quantum efficiency or emission bandwidth of luminescent ions, for example. We have shown in several cases that spectroscopic properties of dopants are not directly related to the average properties of the doped glass, but to their local environment. Indeed, by slightly modifying the silica composition, we succeeded to control the

Figures captions

[

  xlvi]. The fluorescence sensitivity to [Cr] and  p suggests that Cr-ions are located in various host sites, and that several sites are simultaneously selected by an adequate choice of  p (like in Cr(Ge-Al)). It is also suggested that although Al promotes Cr 4+ over Cr 3+ when [Cr] is low, Cr 4+ is also promoted in Ge-modified fibers at high [Cr]. v) The strong decrease of fluorescence from LT to RT is attributed to temperature quenching caused by multiphonon relaxations, like in crystalline materials where the emission drops by typically an order of magnitude from 77 K to 293 K [xxxvii].

  observed LT fluorescence of Cr 4+ is extremely sensitive to glass composition, total Cr concentration and excitation wavelength. Using Aluminum as a glass network modifier has advantages: longer excited state lifetime and broader fluorescence bandwidth than in Germanium-modified silica. A combination of Al and Ge glass modification induces the broadest fluorescence emission in the NIR range, to our knowledge, exhibiting a 550 nmbandwidth. However, increasing the quantum efficiency is now necessary for practical fiber amplifiers and light sources. Further investigations concluded to the necessity of local surrounding TM ions with a different material, i.e. having sensibly different chemical and physical properties compared to pure silica, in order to improve the local site symmetry and hence minimize NRD. Preliminary implementation of this principle was reported recently, concerning Cr 3+ ions in post-heat-treated Ga-modified silica fibers[lv]. When engineering of the local dopant environment will be possible, then practical TM-doped silica-based amplifying devices will be at hand.Phonon interactions / non-radiative relaxations: improvement of Tm 3+ efficiencyThulium-doped fibers have been widely studied in the past few years. Because of Tm 3+ ion rich energy diagram, lasing action and amplification at multiple infrared and visible wavelengths are allowed. Thanks to the possible stimulated emission peaking at 1.47 µm ( 3 H 4 => 3 F 4 , see Fig.9), discovered byAntipenko et al. [lvi], one of the most exciting possibilities of Tm 3+ ion is amplifying optical signal in the S-band (1.47-1.52 µm), in order to increase the available bandwidth for future optical communications. Unfortunately, the upper 3 H 4 level of this transition is very close to the next lower 3 H 5 level so non-radiative de-excitations (NRD) are likely to happen in high phonon energy glass host, causing detrimental gain quenching.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Schematic energy diagram of (a) single energy transfer between two ions, (b) double

Fig. 3 :

 3 Fig.3: Energy scheme for the DET process. Level energies are in cm -1 ; their lifetimes between

Fig. 4 :Fig. 5 :

 45 Fig. 4 : Natural logarithm of measured intensity ratio (R) plotted against the inverse of

Fig. 6 :

 6 Fig. 6:. Background corrected absorption from (left) a Cr(Ge) preform ([Cr] = 1400 ppm) and (right) a Cr(Al) fiber ([Cr] = 40 ppm). Circles: experimental data; Solid lines: adjusted bands to Cr 3+ (left) and Cr 4+ (right) transitions, respectively; and resulting absorption spectra.Assignments are indicated from the ground level Cr 3+ : 4 A 2 or Cr 4+ : 3 A 2 to the indicated excited level, respectively. The Cr 4+ : 3 T 2 level three-fold splitting is due to distorsion from perfect tetrahedral symmetry. The spin-forbidden Cr 3+ : 4 A 2  2 E and Cr 4+ : 3 A 2  1 E transitions are not visible and overlapping with the intense spin-allowed

Fig. 7 :

 7 Fig. 7: Fluorescence spectra: (a) fiber Cr(Al):[Cr] = 40 ppm,  p = 900 nm, T=77 K, (b) preform Cr(Ge-Al): [Cr] = 300 ppm,  p = 673 nm, T=12K.

Fig. 8 :

 8 Fig. 8: Fluorescence decays from Cr(Al) samples,  p = 673 nm, T=12 K: (a)  s ~1100 nm and [Cr]=40 ppm, (b) s ~1100 nm, [Cr]=4000 ppm, and (c)  s ~1400 nm, [Cr]=4000 ppm

Fig. 13 :

 13 Fig. 13 : Room temperature emission spectra of Er-doped preform with (sample A) and without (sample B) Calcium. Samples were excited at 980 nm.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Schematic energy diagram of (a) single energy transfer between two ions, (b) double

Fig. 3 :Fig. 4 :Fig. 5 :

 345 Fig. 3 : Energy scheme for the DET process. Level energies are in cm -1 ; their lifetimes between

Fig. 6 :

 6 Fig. 6:. Background corrected absorption from (left) a Cr(Ge) preform ([Cr] = 1400 ppm) and (right) a Cr(Al) fiber ([Cr] = 40 ppm). Circles: experimental data; Solid lines: adjusted bands to Cr 3+ (left) and Cr 4+ (right) transitions, respectively; and resulting absorption spectra.Assignments are indicated from the ground level Cr 3+ : 4 A 2 or Cr 4+ : 3 A 2 to the indicated excited level, respectively. The Cr 4+ : 3 T 2 level three-fold splitting is due to distorsion from perfect tetrahedral symmetry. The spin-forbidden Cr 3+ : 4 A 2  2 E and Cr 4+ : 3 A 2  1 E transitions are not visible and overlapping with the intense spin-allowed

Fig. 7 :Fig. 8 :Fig. 9 :

 789 Fig. 7: Fluorescence spectra: (a) fiber Cr(Al):[Cr] = 40 ppm,  p = 900 nm, T=77 K, (b) preform Cr(Ge-Al): [Cr] = 300 ppm,  p = 673 nm, T=12K.

Fig. 13 :

 13 Fig. 13 : Room temperature emission spectra of Er-doped preform with (sample A) and without (sample B) Calcium. Samples were excited at 980 nm.

  as Cr 4+ :YAG, have demonstrated very good results as broad-band gain media[xxxvii]. Tentatives with other TM ions, like Ni 2+ in vitroceramics fiber are also promising[xiv]. More recently, a 400-nm emission bandwidth was observed from a fiber whose Cr-doped core was made of Y 2 O 3 :A 2 O 3 :SiO 2 obtained by a rod-in-tube technique using a Cr 4+ :YAG rod as core material and a silica tube as cladding material[xxxviii].Little literature exists on chromium-and other TM-doped vitreous bulk silica, although this issue was addressed in the 70's[xxxix] to improve transmission of silica optical fibers.

Tunability in RE-doped fiber devices is already well established, but limited by shielding of the optically active electronic orbitals of RE ions. Optically active, unshielded orbitals are found in transition metal (TM) ions. Some TM-doped bulk solid-state lasers materials, such

  i) Only Cr3+ and Cr 4+ oxidation states are stabilized. Cr 3+ is favoured by Ge co-doping, and lies in octahedral site symmetry (O), as in other oxide glasses[xlvii]. Cr 4+ is present in all samples. This valency is promoted by Al co-doping or when [Cr] is high, and lies in a distorded YAG (Y 3 Al 5 O 12 )[l]. Glass modifiers like Al induce major spectroscopic changes, even at low concentrations (~1-2 mol%). This would help engineering the Chromium optical properties in silica-based fibers, using possibly alternative modifiers.

tetrahedral site symmetry (C s )

[xlviii,xlix]

. The low-doped Cr(Al) samples contain only Cr 4+ and their absorption spectra are similar to those of aluminate

[xl]

and alumino-silicate glasses

[xli]

and even crystalline

  -24 m 2 is consistent with reported values in other materials, such as ruby [li] and silica glass [xxxix], while  abs (Cr 4+ , 1000 nm)~3.5 x 10 -24 m 2 is lower than in reference crystals for lasers [lii] or saturable absorbers [liii], but consistent with estimated values in alumino-silicate glass [xli].

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decay rates and a modified exponential energy gap law for 4f-4f transitions in rare-earth ions. J. Chem. Phys. 78(9):5317.


Anni Hakanen

Ismael G Yero

Complexity and equivalency of multiset dimension and ID-colorings

Keywords: multiset resolving set, multiset dimension, ID-colorings, ID-number, king grids AMS Subj. Class 05C12, 05C76

published or not. The documents may come

Complexity and equivalency of multiset dimension and ID-colorings

Introduction

The concept of metric dimension in graphs is one of the classical parameters in the area of graph theory. It is understood it has been independently introduced in the decade of 1970 in the two separate works [START_REF] Harary | On the metric dimension of a graph[END_REF] and [START_REF] Slater | Leaves of trees[END_REF], which were aimed to consider identification properties of vertices in a graph. These identification properties were also connected with the Mastermind game in [START_REF] Cáceres | On the metric dimension of Cartesian products of graphs[END_REF], and problems related to pattern recognition and image processing in [START_REF] Melter | Metric bases in digital geometry[END_REF]. After these two seminal works, the research on the topic remained relatively quiet until the first years of the new century, where the number of articles on the topic exploded. From this point on, several theoretical and applied results have been appearing, and nowadays, the metric dimension of graphs is very well studied. It is not our goal to include a lot of references on this fact, and we simply suggest the interested reader to consult the two recent surveys [START_REF] Kuziak | Metric dimension related parameters in graphs: A survey on combinatorial, computational and applied results[END_REF][START_REF] Tillquist | Getting the lay of the land in discrete space: A survey of metric dimension and its applications[END_REF], which have a fairly complete amount of information on metric dimension in graphs and related topics.

where {{•}} represents a multiset. In order to facilitate our exposition, we write {{.}} + i to denote the multiset obtained from {{.}} by adding i to every element of such multiset. The set W is a multiset resolving set for G if the collection of multisets m G (u|S) with u ∈ V (G) are pairwise distinct. The multiset dimension of G, denoted dim ms (G), represents the cardinality of a smallest possible multiset resolving set of G. Multiset resolving sets do not always exist in a given graph. For those graphs G which do not contain any multiset resolving set, the agreement that dim ms (G) = ∞ was taken in [START_REF] Simanjuntak | The multiset dimension of graphs[END_REF]. Some other investigations on the multiset dimension of graphs are [START_REF] Alfarisi | A note on multiset dimension and local multiset dimension of graphs[END_REF][START_REF] Bong | Some properties of the multiset dimension of graphs[END_REF][START_REF] Isariyapalakul | The multibases of symmetric caterpillars[END_REF][START_REF] Khemmani | The characterization of caterpillars with multidimension 3[END_REF][START_REF] Khemmani | The multiresolving sets of graphs with prescribed multisimilar equivalence classes[END_REF]. It is clear that any multiset resolving set is also a resolving set, since the fact that two multisets are different implies that the vectors with the same elements are also different. This means that for any graph G, dim(G) ≤ dim ms (G).

(

) 1
On the other hand, the following concepts were defined in [START_REF] Chartrand | Distance vertex identification in graphs[END_REF]. Consider a connected graph G of diameter d and a set of vertices S ⊂ V (G). Now, for every vertex x ∈ V (G), the code of x with respect to S is the d-vector ⃗ d(x|S) = (a 1 , a 2 , . . . , a d ) where a i , with i ∈ {1, . . . , d} represents the number of vertices in S at distance i from x. If all the codes of vertices of G are pairwise different, then S is called an identification coloring or ID-coloring. Moreover, a graph G that has an ID-coloring is called an ID-graph. In this sense, for any ID-graph G, the cardinality of a smallest ID-coloring is the ID-number of G, denoted by ID(G). Other contributions in this direction are [START_REF] Kono | A note on the identification numbers of caterpillars[END_REF][START_REF] Kono | Vertex identification in grids and prisms[END_REF][START_REF] Kono | Vertex identification in trees[END_REF].

We next show that the two parameters defined above are indeed the same. To this end, we may remark that for an ID-graph G of diameter d with an ID-coloring S, any vertex Based on the equivalence that exists between these two representations, the fact that the codes of vertices of G are pairwise different implies that the collection of multisets are pairwise distinct as well, and vice versa. Consequently, it is clear that a given set S ⊂ V (G) is an ID-coloring of G if and only S is a multiset resolving set.

x ∈ V (G) with code ⃗ d(x|S) = (a 1 , a 2 , . . . , a d ) satisfies d i=1 a i = |S| if x / ∈ S and d i=1 a i = |S| -1 if x ∈ S.
Based on the equivalence above, we conclude the next consequence.

Corollary 2. For any graph G, dim ms (G) = ID(G).

It is then now clear that graphs defined in [START_REF] Chartrand | Distance vertex identification in graphs[END_REF] as ID-graphs are those ones satisfying that dim ms (G) < ∞ according to the terminology from [START_REF] Simanjuntak | The multiset dimension of graphs[END_REF].

Complexity results

This section is centered into considering the proof of the NP-completeness for the following decision problem, which in addition allows to conclude that computing the multiset dimension of graphs is NP-hard, and clearly, based on Theorem 1 and Corollary 2, it means that computing the ID-number of graphs is NP-hard as well. Our proof is somewhat inspired by the proof of the NP-hardness of the outer multiset dimension problem presented in [START_REF] Gil-Pons | Distance-based vertex identification in graphs: the outer multiset dimension[END_REF].

Distinct vertices u and v are called twins if they have the same set of neighbors. It is well known that if two vertices are twins, then each (multiset) resolving set contains at least one of them.

Multiset Dimension

Instance: A graph G = (V, E) and an integer k satisfying 1 ≤ k ≤ |V |. Question: Is dim ms (G) ≤ k? Theorem 3. The Multiset Dimension problem is NP-complete.
Proof. The problem is clearly in NP. We prove the NP-completeness by a reduction from 3-SAT. Consider an arbitrary input to 3-SAT, that is, a formula F with n variables and m clauses, which does not have a clause containing both the positive and the negative literals of the same variable. Let x 1 , x 2 , . . . , x n be the variables, and let C 1 , C 2 , . . . , C m be the clauses of F . We next construct a connected graph G based on this formula F . To this end, we use the following gadgets.

For each variable x i we construct a variable gadget as follows (see Figure 1(a)). Vertices T i , F i are the "true" and "false" ends of the gadget. The gadget is attached to the rest of the graph only through these vertices.

Vertices

a 1 i , a 2 i , b 1 i , b 2
i represent the value of the variable x i , that is, a 1 i and a 2 i will be used to represent that variable x i is true, and b 1 i and b 2 i that it is false. The vertices a 1 i and b 1 i are adjacent, and so are the vertices a 2 i and b 2 i . Additionally, the vertices a 1 i and a 2 i are adjacent to T i and the vertices b 1 i and b 2 i are adjacent to F i .

P t i = d 1 i d 2 i • • • d t i
i is a path such that d 1 i is adjacent to T i and F i , while d t i i is adjacent to the two vertices e 1 i and e 2 i . Notice that the vertices e 1 i and e 2 i are twins, and so, each multiset resolving set of G must contain at least one of them.

For each clause C j we construct a clause gadget as follows (see Figure 1(b)).

The vertices c 1 j , c 2 j and c 3 j form a path. The vertices c 1 j and c 3 j will be helpful in determining whether the clause C j is satisfied.

P s j = f 1 j f 2 j • • • f s j j is a path such that f 1 j is adjacent to c 2 j , while f s j
j is adjacent to the two vertices g 1 j and g 2 j . The vertices g 1 j and g 2 j are twins, and so, each multiset resolving set of G must contain at least one of them.

The orders of the paths P t i and P s j must be pairwise distinct in order for the multiset representations to be different and our reduction to work. Let t i = 5(i+1) for all i ∈ {1, . . . , n} and s j = 5(n + j + 1) for all j ∈ {1, . . . , m}. The sum of all the vertices in the variable and clause gadgets is clearly polynomial in terms of n + m.

The variable and clause gadgets are connected in the following way in order to construct our graph G.

Vertices c 1 j are adjacent to vertices T i , F i for all j and i. If a variable x i does not appear in a clause C j , then the vertices T i , F i are adjacent to c 3 j . If a variable x i appears as a positive literal in a clause C j , then the vertex F i is adjacent to c 3 j . If a variable x i appears as a negative literal in a clause C j , then the vertex T i is adjacent to c 3 j . Observe that G is connected and its order is polynomial in terms of the quantity of variables and clauses of the 3-SAT instance. We shall show that F is satisfiable if and only if dim ms (G) = 2m + n. To this end, we proceed with a series of claims that will complete our whole reduction.

a 1 i b 1 i a 2 i b 2 i T i F i d 1 i d 2 i . . .
Claim 1. We have dim ms (G) ≥ 2n + m.

Proof of Claim 1. Let S be a multiset basis of G. The vertices e 1

i and e 2 i are twins, and thus S contains at least one of them for every i ∈ {1, . . . , n}. Similarly, the vertices g 1 j and g 2 j are twins and at least one of them is in S for every j ∈ {1, . . . , m}. Finally, in order to have distinct multiset representation for the vertices a 1 i , a 2 i , b 1 i and b 2 i , at least one of these four vertices must be in S. Thus, we have dim

ms (G) = |S| ≥ 2n + m. ♦ Claim 2. If F is satisfiable, then dim ms (G) = 2n + m.
Proof of Claim 2. Consider a satisfying assignment for F and construct a set S * containing 2n + m vertices as next described.

For each i ∈ {1, . . . , n}, we add the vertex e 1 i to S * . For each j ∈ {1, . . . , m}, we add the vertex g 1 j to S * . For each variable x i , if x i = true, then we add the vertex a 1 i to S * , otherwise if x i = false, then we add the vertex b 1 i to S * .

We will show that the set S * is a multiset resolving set of G. We denote by S * x i and S * C j the vertices of the set S * that are not in the gadget of x i and C j , respectively. We will first express the multiset representations of the vertices in the variable gadgets with the help of the multiset representation of T i with respect to S * x i . Since the vertices c 1 j are adjacent to all T i and F i , the distance from T i to all T i ′ and

F i ′ for i ′ ̸ = i is 2. Now, we have m(T i |S * x i ) = {{ 3, | for each i ′ ∈ {1, . . . , n}, i ′ ̸ = i t i ′ + 3, | for each i ′ ∈ {1, . . . , n}, i ′ ̸ = i s j + 3, | for each j ∈ {1, . . . , m}}}.
For instance, the distance between the vertex T i and e 1 i ′ equals t i ′ +3, because one shortest path between them is

T i c 1 1 T i ′ d 1 i ′ • • • d t i ′ i ′ e 1 i ′ .
Other cases are deduced similarly from the construction of the graph G. Now, we can write the multiset representations of the vertices in the variable gadget of x i as follows.

m

(T i |S * ) = m(T i |S * x i ) ∪ {{t i + 1, y}}, where y = 1 when a 1 i ∈ S * and y = 2 when b 1 i ∈ S * . m(F i |S * ) = m(T i |S * x i ) ∪ {{t i + 1, y}}, where y = 1 when b 1 i ∈ S * and y = 2 when a 1 i ∈ S * . m(d h i |S * ) = (m(T i |S * x i ) + h) ∪ {{t i -h + 1, h + 1}} for all h ∈ {1, . . . , t i }. m(e h i |S * ) = (m(T i |S * x i ) + t i + 1) ∪ {{t i + 2, y}}, where y = 0 when e h i = e 1 i and y = 2 when e h i = e 2 i . m(a h i |S * ) = (m(T i |S * x i ) + 1) ∪ {{t i + 2, y}},
where when a 1 i ∈ S * , we have y = 0 for a 1 i and y = 2 for a 2 i , and when b 1 i ∈ S * , we have y = 1 for a i 1 and y = 3 for As for the vertices of the clause gadgets, we will express them with an auxiliary representation as well. To that end, observe that m(c

a 2 i . m(b h i |S * ) = (m(T i |S * x i ) + 1) ∪ {{t i + 2,
1 j |S * C j ) = {{ 2, | for each i ∈ {1, . . . , n} t i + 2, | for each i ∈ {1, . . . , n} s j ′ + 4, | for each j ′ ∈ {1, . . . , m}, j ′ ̸ = j}}.
We then write the multiset representations of the vertices of the clause gadget of C j other than c 3 j as follows. m(c

1 j |S * ) = m(c 1 j |S * C j ) ∪ {{s j + 2}}. m(c 2 j |S * ) = (m(c 1 j |S * C j ) + 1) ∪ {{s j + 1}}. m(f h j |S * ) = (m(c 1 j |S * C j ) + h + 1) ∪ {{s j -h + 1}} for all h ∈ {1, . . . , s j }. m(g h j |S * ) = (m(c 1 j |S * C j ) + s j + 2) ∪ {{y}}
, where y = 0 when g h j = g 1 j and y = 2 when g h j = g 2 j . Since the values of t i and s j are large and distinct enough (recall that t i = 5(i + 1) and s j = 5(n + j + 1)), the multiset representations of the vertices in the variable and clause gadgets (other than c 3 j ) are pairwise distinct. Indeed, notice that the values t i ′ + 3 and s j + 3 form a pattern to the multiset representations of a vertex of a variable gadget which is easy to distinguish from the corresponding pattern of t i +2 and s j ′ +4 of a vertex from a clause gadget. Thus, we readily observe that the multiset representations of the vertices in a variable gadget are distinct from those of the vertices of clause gadgets. These patterns, or the anomalies present in them, to be more precise, are also the reason why vertices in two different variable gadgets (or clause gadgets) have distinct multiset representations. Indeed, there is a "gap" in this pattern where t i + 3 should be for all vertices of the gadget of x i . Thus, the multiset representations of the vertices within the same variable or clause gadget are distinct.

Let us then consider the vertices c 3 j . Similarly to the vertex c 1 j , the multiset representation of c 3 j contains s j + 2 once, t i + 2 for each i ∈ {1, . . . , n}, and s j ′ + 2 for each j ′ ∈ {1, . . . , m}, j ′ ̸ = j. However, the distance from c 3 j to the vertices a 1 i and b 1 i is 2 or 3 depending on whether the variable x i appears in the clause C j and in which form (positive or negative), and whether x i = true or x i = false according to the truth assignment. More precisely, if the variable x i does not appear in the clause

C j , then d(c 3 j , a 1 i ) = d(c 3 j , b 1 i ) = 2.
If x i appears in C j but the clause C j is not satisfied due to x i (note that here C j can be satisfied but due to the truth value of some other variable x i ′ ), then the distance from c 3 i to whichever of a 1 i and b 1 i is in S * is again 2. However, if C j is satisfied due to x i , the distance from c 3 j to whichever of a 1 i and b 1 i is in S * is 3. (For example, if x i appears as a positive literal in C j , the edge c 3 j F i is present in G whereas c 3 j T i is not. Thus, we have d(c 3 j , a 1 i ) = 3 and d(c 3 j , b 1 i ) = 2. Now, if the truth assignment of x i leads to C j being satisfied, we have x i = true and a 1 i ∈ S * . Therefore, the multiset representation of c 3 j contains 3 due to a 1 i .) Since the set S * is constructed using a truth assignment that satisfies F , there is at least one 3 in the multiset representation of c 3 j . Thus, the multiset representation of c 3 j is almost the same as the multiset representation of c 1 j except that at least one 2 (in m(c 1 j |S * )) is swapped to 3. Thus, c 3 j and c 1 j have distinct multiset representations. Furthermore, based on the arguments concerning the multiset representations of the other vertices of G, it is clear that each c 3 j has a distinct multiset representation compared to all other vertices of G. Consequently, the set S * is a multiset resolving set of G, and the claim holds due to Claim 1.

♦ Claim 3. If dim ms (G) = 2n + m, then F is satisfiable.
Proof of Claim 3. Let S be a multiset basis of G. By the arguments in the proof of Claim 1, the set S must contain exactly one of the two vertices e 1 i or e 2 i for every i ∈ {1, . . . , n}; exactly one of the two vertices g 1 j or g 2 j for every j ∈ {1, . . . , m}; and exactly one of the vertices a 1 i , a 2

i , b 1 i or b 2 i for every i ∈ {1, . . . , n}. By the same arguments as at the end of the proof of Claim 2, the vertices c 1 j and c 3 j must have distinct multiset representations due to some a h i or b h i . Thus, the truth assignment where x i = true if a 1 i or a 2 i is in S, and

x i = false if b 1 i or b 2 i is in S for all i ∈ {1, . . . , n} satisfies F . ♦
This completes the reduction from 3-SAT to the Multiset Dimension problem.

The king grid

Based on the NP-completeness reduction made in the proof of Theorem 3, it is then desirable to consider the multiset dimension (or ID-number) of some non-trivial families of graphs. In connection with this, in this section we consider the strong product of a path P n with itself, also known as the king grid. The graph G ⊠ H is the strong product of G and H. The vertex set of G ⊠ H is the set

V (G) × V (H) = {(u, v) | u ∈ V (G), v ∈ V (H)}. Two vertices (g, h), (g ′ , h ′ ) ∈ V (G ⊠ H) are adjacent if g = g ′
and h is adjacent to h ′ in H; or g is adjacent to g ′ in G and h = h ′ ; or g is adjacent to g ′ in G and h is adjacent to h ′ in H. We write V (P n ) = {1, . . . , n} so that the vertices of P n ⊠ P n correspond to the coordinates of the Z 2 lattice. Moreover, for a vertex (i, j) ∈ V (P n ⊠P n ) and an integer q ≥ 1, by D q (i, j) we mean the set of vertices in V (P n ⊠P n ) at distance q from (i, j). Notice that such set D q (i, j) represents a kind of (not necessarily whole) "border" of a subgraph of P n ⊠ P n isomorphic to the strong product of two paths. See Figure 2 for two representative examples. Remark 4. We have dim ms (P 2 ⊠ P 2 ) = dim ms (P 3 ⊠ P 3 ) = ∞, since both of these graphs are of diameter at most 2, and such graphs have no multiset resolving sets [START_REF] Simanjuntak | The multiset dimension of graphs[END_REF]. As for the case n = 4, the set {(1, 1), (2, 1), (4, 1), (1, 3), (2, 3), (2, 4)} is a multiset resolving set of P 4 ⊠ P 4 . We have checked with an exhaustive computer search that no smaller multiset resolving sets exist for this graph, and thus dim ms (P 4 ⊠ P 4 ) = 6

We begin our exposition of the larger king grids with the two smallest cases, and further on proceed with the general case.

Proposition 5. We have dim ms (P 5 ⊠ P 5 ) = dim ms (P 6 ⊠ P 6 ) = 4.

Proof. It is known from [START_REF] Simanjuntak | The multiset dimension of graphs[END_REF] and [START_REF] Chartrand | Distance vertex identification in graphs[END_REF] that no graph has a multiset resolving set of cardinality 2. Also, the only graph that has a multiset resolving set consisting of only one element is the path graph P n . Thus, dim ms (P 5 ⊠ P 5 ) ≥ 3 and dim ms (P 6 ⊠ P 6 ) ≥ 3. We first prove that dim ms (P 5 ⊠P 5 ) ̸ = 3 by a simple counting argument. Suppose that S is a multiset resolving set of P 5 ⊠ P 5 such that |S| = 3. The maximum number of distinct multiset representations that do not contain 0 is 6 3 = 20. Since |V (G) \ S| = 22, some vertices of P 5 ⊠ P 5 have the same multiset representations, a contradiction. Thus, dim ms (P 5 ⊠ P 5 ) > 3, and the first equality follows since the set S = {(1, 1), (2, 1), (5, 1), (1, 5)} is a multiset resolving set of P 5 ⊠ P 5 . The sets S along with the multiset representations is illustrated in Figure 3.

On the other hand, observe that the set S ′ = {(2, 1), (2, 2), (6, 1), (1, 6)} is a multiset resolving set of P 6 ⊠ P 6 , as shown in Figure 3, throughout the multiset representations of each vertex with respect to S ′ , being pairwise different. Thus dim ms (P 6 ⊠ P 6 ) ≤ 4. Now, in contrast to the case of P 5 ⊠ P 5 , to prove that dim ms (P 6 ⊠ P 6 ) ̸ = 3, the counting argument used does not directly work. That is, if we suppose S ′′ is a multiset resolving set of P 6 ⊠ P 6 such that |S| = 3, then the maximum number of distinct multiset representations that do not contain 0 is required. To this end, assume there is a vertex (α, β) ∈ V (P 6 ⊠ P 6 ) such that it has multiset representation {{1, 1, a}} for some a ∈ {1, . . . , 5}. Let (i, j), (i ′ , j ′ ) be two neighbors of (α, β) in S ′′ (note that (i, j), (i ′ , j ′ ) are at distance at most two). Hence, the third vertex (i ′′ , j ′′ ) of S ′′ must be in the set D a (α, β), namely, S ′′ = {(i, j), (i ′ , j ′ ), (i ′′ , j ′′ )}.

It is now just a matter of checking all the possibilities that can occur between the two vertices (i, j), (i ′ , j ′ ) and the third vertex (i ′′ , j ′′ ), to observe that one can always find two vertices that have the same multiset representation with respect to S ′′ . In order to avoid a lengthy and time consuming case analysis, we have simply checked this by computer. Thus, the multisets representations {{1, 1, a}} are not possible for every a ∈ {1, . . . , 5} with respect to the set S ′′ , and there are 5 of them. But then this means we have a total amount of 7 3 -5 = 30 possible distinct multiset representations that do not contain 0. However, there are |V (P 6 ⊠ P 6 ) \ S ′′ | = 33 vertices, which is a contradiction. Therefore, dim ms (P 6 ⊠ P 6 ) ̸ = 3, and the equality dim ms (P 6 ⊠ P 6 ) = 4 follows.

From now on, in order to facilitate the exposition, by a row or a column in P n ⊠P n we mean the path induced by the vertices (1, j), (2, j) . . . , (n, j) or (i, 1), (i, 2) . . . , (i, n), respectively, for any i, j ∈ {1, . . . , n}.

Theorem 6. If n ≥ 7 is an integer, then 3 ≤ dim ms (P n ⊠ P n ) ≤ 4.
Proof. The lower bound follows from the fact that any graph different from a path has multiset dimension at least 3, or by using inequality (1), since dim(P n ⊠ P n ) = 3 (see [START_REF] Barragán-Ramírez | The local metric dimension of strong product graphs[END_REF][START_REF] Rodríguez-Velázquez | The metric dimension of strong product graphs[END_REF]). To show the upper bound we use an induction procedure that separately works for odd and even values of n.

Case 1: n ≥ 7 is odd. Assume that the set S n-2 = {(1, 1), (2, 1), (n -2, 1), (1, n -2)} is a multiset resolving set of P n-2 ⊠ P n-2 . This is true for n = 7 according to Proposition 5, which shows the base case of the induction process. We will show that the set S n = {(1, 1), (2, 1), (n, 1), (1, n)} is a multiset resolving set of P n ⊠ P n .

Denote F = {(x, y) ∈ V (P n ⊠ P n ) | x ∈ {1, n} or y ∈ {1, n}} and I = V (P n ⊠ P n ) \ F (see Figure 4(a)). Observe that each vertex in F has n -1 in its multiset representation with respect to S n , whereas the vertices in I do not. Thus, vertices from F and I clearly have multiset representations distinct from one another.

Consider the vertices in I. The graph P n ⊠ P n can be viewed as a graph constructed from P n-2 ⊠ P n-2 by adding an additional row or column of vertices to all four sides of the graph. The set S n can then be obtained by moving the elements of S n-2 diagonally away from the middle. Thus, for each (v [START_REF] Bong | Some properties of the multiset dimension of graphs[END_REF][START_REF] Barragán-Ramírez | The local metric dimension of strong product graphs[END_REF] have distinct multiset representations with respect to each other. Moreover, the vertex (2, 2) is the only vertex in I that is adjacent to two elements of the set S 2 . The vertex (3, 2) is adjacent to one element of the set S n , and so are two other vertices in I. However, neither of these vertices has the distance 2 in their multiset representations, which the vertex (3, 2) does have. Thus, all vertices in I have pairwise distinct multiset representations with respect to S n .

1 , v 2 ) ∈ I \ {(2, 2), (3, 2)}, we have m((v 1 , v 2 )|S n ) = m((v 1 -1, v 2 -1)|S n-2 ) + 1. Since S n-2 is a multiset resolving set of P n-2 ⊠ P n-2 , all vertices in I \ {(2, 2), (3, 2)} have distinct multiset representations with respect to S n in P n ⊠P n . Since m((2, 2)|S n ) = {{1, 1, n- 2, n -2}} and m((3, 2)|S n ) = {{1, 2, n -3, n -2}}, the vertices (2, 2) and
Consider then the vertices in F . The multiset representations of these vertices are as follows.

For vertices on the top row, i.e. (i, n) where i ∈ {1, . . . , n}, we have m

((i, n)|S n ) = {{i -1, n -1, n -1, n -1}}.
For vertices on the right column, excluding the top corner, i.e. (n, i)

where i ∈ {1, . . . , n -1}, we have m((n, i)|S n ) = {{i -1, n -2, n -1, n -1}}.
For vertices on the left column, excluding top and bottom corners, i.e. (1, i) where i ∈ {2, . . . , n -1}, we have m((

1, i)|S n ) = {{i -1, i -1, n -i -2, n -1}}.
For vertices on the bottom row, excluding both corners, i.e. (i, 1), where i ∈ {2, . . . , n -1}, we have m((i, 1

)|S n ) = {{i -1, i -2, n -i -2, n -1}}.
For the bottom left corner (1, 1), we have m((1, 1

)|S n ) = {{0, 1, n -1, n 1}}.
Since the vertices in the top row are the only ones to have (at least) three (n -1)'s in their multiset representations, the vertices on the top row have distinct representations from the other vertices of F . Moreover, the multiset representations of the vertices on the top row are pairwise distinct. Similarly, from the vertices left to be considered, the vertices in the right column are the only vertices that have exactly two (n -1)'s in their representations (other than [START_REF] Alfarisi | A note on multiset dimension and local multiset dimension of graphs[END_REF][START_REF] Alfarisi | A note on multiset dimension and local multiset dimension of graphs[END_REF], but that has a distinct multiset representation due to the 0). Thus it is clear that the vertices in the right column have distinct multiset representations with respect to each other and other vertices in F . The vertices in the left column have pairwise distinct representations with respect to each other, and the same holds also for vertices in the bottom row. The only thing left to show is that two vertices, one in the left column and the other in the bottom row, cannot have the same multiset representations.

To that end, suppose to the contrary that (i, 1) and (1, j) have the same multiset representation for some i, j ∈ {2, . . . , n -1}. Since j -1 appears twice in the multiset representation of (1, j), we must have i -

1 = n -i -2 or i -2 = n -i -2. Since n is odd, we have i -1 = n -i -2 and i -1 = n-3 2 . Now, m((i, 1)|S n ) = {{ n-3 2 , n-3 2 , n-3 2 -1, n -1}}. This implies that j -1 = n-3 2 , but now m((1, j)|S n ) = {{ n-3 2 , n-3 2 , n-3 2 , n -1}} ̸ = m((i, 1)
|S n ), a contradiction. Thus, all vertices of F have pairwise distinct multiset representations with respect to S n .

As a consequence, we obtain that S n is a multiset resolving set as claimed, and so, dim ms (P n ⊠ P n ) ≤ 4 in this case. Case 2: n ≥ 8 is even. Assume that the set S n-1 = {(1, 1), (2, 1), (n -1, 1), (1, n -1)} is a multiset resolving set of P n-1 ⊠ P n-1 . If n = 8, then by the Case 1 we know that S 7 is a multiset resolving set of P 7 ⊠ P 7 , which shows the base case. We will show that the set S n = {(2, 2), (2, 3), (n, 2), (2, n)} is a multiset resolving set of P n ⊠ P n . We now denote F = {(x, y) ∈ V (P n ⊠ P n ) | x = 1 or y = 1} and I = V (P n ⊠ P n ) \ F (see Figure 4(b)). The vertices in I clearly have pairwise distinct multiset representations as m(v|S n ) = m(v|S n-1 ) for all v ∈ I, and S n-1 is a multiset resolving set of P n-1 ⊠ P n-1 . Also, all the vertices in F have n -1 in their multiset representations with respect to S n , whereas vertices in I do not. Thus, the multiset representation of a vertex in F is always distinct from that of a vertex of I.

We will show next that the multiset representations of vertices of F are pairwise distinct. To that end, we first consider the vertices that are adjacent to some element of S n . Their multiset representations are the following:

m((1, 1)|S n ) = {{1, 2, n -1, n -1}}, m((1, n)|S n ) = {{1, n -2, n -2, n -1}}, m((1, 2)|S n ) = {{1, 2, n -2, n -1}}, m((1, n -1)|S n ) = {{1, n -3, n -3, n -1}}, m((1, 3)|S n ) = {{1, 2, n -3, n -1}}, m((n, 1)|S n ) = {{1, n -3, n -2, n -1}}, m((4, 1)|S n ) = {{1, 2, n -4, n -1}}, m((n -1, 1)|S n ) = {{1, n -4, n -3, n -1}}, m((2, 1)|S n ) = {{1, 1, n -2, n -1}}, m((3, 1)|S n ) = {{1, 1, n -3, n -1}}.
Since n ≥ 8, we have n -4 ̸ = 2 and all these multiset representations are pairwise distinct. Moreover, since these vertices are the only vertices in F that have the distance 1 in their multiset representations, these multiset representations are distinct from the multiset representation of other vertices in F as well. Let us then consider the rest of the vertices in F . The vertices (i, 1) where i ∈ {5, . . . , n -2} have multiset representations of the form {{i -2, i -3, n -i -2, n -1}}, and these representations are pairwise distinct. Similarly, the vertices (1, j) where j ∈ {4, . . . , n -2} have multiset representations of the form {{j -2, j -2, n -j -2, n -1}}, and these representations are clearly pairwise distinct. Suppose then that (i, 1) and (1, j) have the same multiset representation for some i ∈ {5, . . . , n -2} and j ∈ {4, . . . , n -2}. As in the proof for odd n, the distance j -2 appears twice in the multiset representation of (1, j). This implies that i

-2 = n -i -2 or i -3 = n -i -2.
As n is even, we have i -2 = n -i -2, and thus i -

2 = n-4 2 . Now, m((i, 1)|S n ) = {{ n-4 2 , n-4 2 , n-4 2 -1, n -1}}. This implies that j -2 = n-4 2 . However, now m((1, j)|S n ) = {{ n-4 2 , n-4 2 , n-4 2 , n -1}} ̸ = m((i, 1)|S n ), a contradiction.
Thus, all vertices in F have pairwise distinct multiset representations with respect to S n .

Therefore, we again conclude that S n is a multiset resolving set in this situation, and so, dim ms (P n ⊠ P n ) ≤ 4 follows as well, which completes the proof.

Strong products involving a complete graph

Several graphs with infinite multiset dimension (or equivalently that are not ID-graphs) are already known from the seminal works [START_REF] Chartrand | Distance vertex identification in graphs[END_REF][START_REF] Simanjuntak | The multiset dimension of graphs[END_REF], in their corresponding terminologies. For instance, it is known from these mentioned works that for a given graph G of diameter two, dim ms (G ⊠ K n ) < ∞ if and only G is P 3 . From this result we can identify a lot of interesting and non-trivial families of graphs like for instance the join of two non-complete graphs, the Cartesian or direct product of two complete graphs, and the Kneser graph K(n, 2), among others, having infinite multiset dimension.

We next describe some other graphs with diameter larger than two that have infinite multiset dimension. Specifically, we consider the case of strong product graphs G ⊠ H when H is a complete graph. We first need the following definition from [START_REF] Klavzar | Further contributions on the outer multiset dimension of graphs[END_REF]. A graph G is called a multiset distance irregular graph if for any two vertices x, y ∈ V (G) it follows that m(x|V (G)) ̸ = m(y|V (G)). A example of a multiset distance irregular graph is for instance a tree obtained from a star S n with leaves v 0 , . . . v n-1 (n ≥ 3) and center x, by subdividing i times the edge xv i for every i ∈ {0 . . . , n -1}.

Theorem 7. Let G be a graph and let n ≥ 2 be an integer. We have dim ms (G ⊠ K n ) < ∞ if and only if n = 2 and G is a multiset distance irregular graph.

Proof. Notice that the n vertices of each copy of K n are twins, that is, they have the same closed neighborhood. Hence, if S is a multiset resolving set of G ⊠ K n , then at least n -1 vertices from each copy of K n must be in S. If n ≥ 3, then from each copy of K n there are at least two vertices in S. But then, these two vertices have the same multiset representation, which is not possible. Consequently, we deduce that G ⊠ K n does not have multiset resolving sets when n ≥ 3 and so, dim ms (G ⊠ K n ) = ∞ in this case.

Assume next that n = 2. By the same reasons as above, at least one vertex from each copy of K 2 must be in S. Thus dim ms (G ⊠ K 2 ) ≥ |V (G)|. Now, assume G is multiset distance irregular, and consider a set X of vertices containing one whole copy of G. Since the multisets representations of any two vertices x, y ∈ V (G) satisfy that m(x|V (G)) ̸ = m(y|V (G)), we deduce that X is a multiset resolving set, and so dim ms (G ⊠ K n ) ≤ |V (G)|, which gives the equality dim ms (G ⊠ K n ) = |V (G)| < ∞.

Assume now that dim ms (G ⊠ K n ) < ∞. Clearly, n = 2, for otherwise we get a contradiction. Also, we readily see that every multiset resolving set has nonempty intersection with every copy of K 2 in G ⊠ K 2 and that dim ms (G ⊠ K 2 ) ≥ |V (G)|. Let X ′ be a multiset basis of G ⊠ K 2 . If X ′ contains both vertices of one copy of K 2 , then these two vertices have the same multiset representation as they are twins. Thus, X ′ contains exactly one vertex from each copy of K 2 . If G is not multiset distance irregular, then there are two vertices x, y ∈ V (G) such that m(x|V (G)) = m(y|V (G)). Now, the vertices in the copies of K 2 corresponding to x and y that are not in X ′ have the same multiset representation, a contradiction. Therefore, G is multiset distance irregular, and the proof is completed.

Concluding remarks

This work firstly shows that two metric parameters represent the same in graph theory, that is, multiset dimension and ID-colorings are the same. We have also considered some computational and combinatorial problems on this parameter. As a consequence of the study a number of possible future research lines have been detected. We next remark a few that could be of interest from our humble opinion.

We have proved that finding the multiset dimension of graphs is in general NP-hard. However, not much is known on special classes of graphs. Does finding such parameters remain NP-hard even when restricted to trees, chordal graphs or planar graphs?

The multiset dimension of the king grid has been bounded above by 4. Is it true that dim ms (P n ⊠ P n ) = 4 for any n ≥ 7? In addition, this result gives a first step into considering the multiset dimension of the strong product graphs in general. Moreover, the study of such parameter in some other related graph products is worthwhile as well.

We have related the multiset dimension of graphs with multiset distance irregular graphs. Does such graphs play a significant role in some investigations on the multiset dimension of general graphs or at least in some product related structures?

The notion of identification spectrum of an ID-graph G was introduced in [START_REF] Chartrand | Distance vertex identification in graphs[END_REF], which can be understood as the set of positive integers r for which there is a multiset resolving set of cardinality r. In this sense, since we have proved that dim ms (P n ⊠ P n ) ≤ 4 (equivalently P n ⊠ P n is an ID-graph) for any n ≥ 7, we wonder which is the identification spectrum of P n ⊠ P n . Characterize all the graphs G for which dim(G) = dim ms (G).

Theorem 1 .

 1 Let G be a graph of diameter d. Then S ⊂ V (G) is an ID-coloring for G if and only if S is a multiset resolving set for G.Proof. The result follows directly from the following fact. Let S ⊂ V (G) and let x ∈ V (G). Consider the multiset representation m G (x|S) = {{d G (x, w 1 ), . . . , d G (x, w t )}}. From m G (x|S) we construct the vector ⃗ d(x|S) = (a 1 , a 2 , . . . , a d ) as follows. Each a i equals the number of elements in m G (x|S) with value i for any i ∈ {1, . . . , d}. On the contrary, if we have the vector ⃗ d(x|S) = (a 1 , a 2 , . . . , a d ), then the multiset representation m G (x|S) can be obtained in the following way. First, if d i=1 a i = |S| -1, then x ∈ S and we need to add the value zero (0) to m G (x|S). Otherwise, if d i=1 a i = |S|, then x / ∈ S, and the value zero (0) does not belong to m G (x|S). Now, in both cases, for every a i ∈ ⃗ d(x|S) such that a i ̸ = 0, we add to m G (x|S) a i elements equal to i.

d t i i e 1 i e 2 i 2 j

 22 (a) The variable gadget for x i . (b) The clause gadget for C j .

Figure 1 :

 1 Figure 1: The variables and clause gadgets used in the reduction.

  y}}, where when a 1 i ∈ S * , we have y = 1 for b 1 i and y = 3 for b 2 i , and when b 1 i ∈ S * , we have y = 0 for b i 1 and y = 2 for b 2 i .

Figure 2 :

 2 Figure 2: The graph P 6 ⊠ P 6 with the sets D 2 (2, 2) and D 2 (3, 3), respectively illustrated in black. The vertices (2, 2) and (3, 3) appear in red color.

Figure 3 :

 3 Figure 3: The graphs P 5 ⊠ P 5 and P 6 ⊠ P 6 with the sets S and S ′ , respectively illustrated in black. The four digits below each vertex are the distances in the multiset representation sorted in ascending order.

8 Figure 4 :

 84 Figure 4: Vertices within the dashed line are the vertices in I, and the vertices outside the dashed line are the vertices in F . The black vertices are the elements of the set S n .

Corollary 8 .

 8 For any multiset distance irregular graph G, dim ms (G ⊠ K 2 ) = |V (G)|.

Acknowledgements

Anni Hakanen has been supported by the Jenny and Antti Wihuri Foundation, and also partially supported by the ANR project GRALMECO (ANR-21-CE48-0004-01) and Academy of Finland grant number 338797. Ismael G. Yero has been partially supported by the Spanish Ministry of Science and Innovation through the grant PID2019-105824GB-I00.

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Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.