C Girod

David Leboeuf

A Demuer

G Seyfarth

S Imajo

K Kindo

Y Kohama

M Lizaire

A Legros

A Gourgout

H Takagi

T Kurosawa

M Oda

N Momono

J Chang

S Ono

G.-Q Zheng

C Marcenat

L Taillefer
email: louis.taillefer@usherbrooke.ca

T Klein

Normal state specific heat in the cuprates La 2-x Sr x CuO 4 and Bi 2+y Sr 2-x-y La x CuO 6+δ near the critical point of the pseudogap phase

teaching and research institutions in France or abroad, or from public or private research centers.   Normal state specific heat in the cuprate superconductors La 2 -x Sr x CuO 4 and Bi 2 + y Sr 2 -x -y La x CuO 6 + δ near the critical point of the pseudogap phase

I. INTRODUCTION

Unravelling the mystery of high temperature superconductivity remains a fundamental issue in modern solid state physics. A central question is the nature of the enigmatic pseudogap phase which appears below a temperature T and a critical hole concentration (doping) p . Well above p , the Fermi surface consists of a large quasi-2D cylinder (see for instance [1,2] in Tl 2 Ba 2 CuO 6+δ ), the measured carrier concentration is equal to n H = 1 + p (per CuO 2 plane) and the Sommerfeld coefficient is on the order of 5 mJmol -1 K -2 [3][4][5][6]. On the other hand, for p ≤ p , ARPES studies show that the Fermi surface breaks into small nodal 'Fermi arcs' [7] and Hall effect measurements then indicate that the carrier concentration drops to n H = p in YBa 2 Cu 3 O y (YBCO) [8], Nd 0.4 La 1.6-x Sr x CuO 4 (Nd-LSCO) [9] and Bi 2+y Sr 2-x-y La x CuO 6+δ (Bi2201) [10,11].

Specific heat measurements in the normal state of Nd-LSCO and Eu 0.2 La 1.8-x Sr x CuO 4 (Eu-LSCO) [4] recently showed that the electronic contribution to the specific heat, C e /T , actually displays a pronounced peak as function of doping at p ∼ p . Moreover, for p close p , the electronic specific heat displays a logarithmic temperature dependence : C e /T = Bln(T 0 /T ) [4]. Both behaviors are typical thermodynamic signatures of quantum criticality. It is important to investigate whether those characteristic features are also present -or not -in other cuprates. We hence report here a study of the temperature and doping dependence of the electronic specific heat C e in La 2-x Sr x CuO 4 (LSCO) and Bi2201 single crystals.

In LSCO, large C e /T values (on the order of 15 mJmol -1 K -2 at 2 K, are measured in the vicinity of the onset of the pseudogap phase. However, in contrast 3). The small overshoot observed for p ∼ 0.24 is most probably reminiscent of the superconducting transition in presence of strong fluctuations. The shaded (colored) boxes indicate the locus of the previously estimated Hc2(0) values [13,[START_REF] Wang | [END_REF]. Open circles are low field data extracted from [6]. As seen, (C -C ph )/T increases with p, reaching ∼ 15-16 mJmol -1 K -2 in the normal state for p ∼ 0.22-0.25 at T ∼ 2 K (see also Fig. 2).

to previous measurements in Nd/Eu-LSCO [4], C e /T remains large over an extended doping range, confirming the former indication for the presence of a broad maximum in the doping dependence of C e /T observed in Zn substituted samples [12]. Despite the presence of a large (hyperfine) Schottky contribution, we also show that a ln(1/T ) contribution has to be introduced in order to fit the temperature dependence of C e /T . Similarly, a clear ln(1/T ) contribution to C e /T and concomitant large C e /T values (on the order of 13 mJmol -1 K -2 at 0.65 K) are observed in BSCO for p ∼ p , hence confirming the universality of those features. Very similar values of the B coefficient are observed in all compounds (close to p ) and the influence of the magnetic field on T 0 will be discussed.

II. METHODS

The specific heat of seven LSCO single crystals with 0 ≤ p ≤ 0.25 and four Bi2201 single crystals with doping contents close to the onset of the pseudogap phase has been measured by AC micro-calorimetry down to ≈ 0.5 K and up to 35 T. Sample LSCO#3 and LSCO#4 are the same samples than measured in references [13,14]. A transport study of sample Bi2201#1 and Bi2201#4 (labelled OD10K and OD18K, respectively) has recently been performed in [11]. The temperature T * was determined by the temperature dependence of the Knight shift for Bi2201 #2 and #3 [15] The heat capacity has been measured by a modulation technique where a periodically modulated heating power P ac is applied at a frequency 2ω. Recording the amplitude of the induced temperature oscillations |T ac | and its thermal phase shift φ relative to the power allows to calculate the heat capacity : C p = P ac sin(-φ)/2ω|T ac |. A miniature CERNOX resistive chip has been split into two parts and attached to a small copper ring with PtW(7 %) wires. The first half (R H ) was then used as an electrical eater (P ac = R H i 2 ac (ω)/2) and the second (R T ) was used to record the temperature oscillations (V ac (2ω) = (dR T /dT )T ac (2ω)i DC , where i DC is a DC reading current). In order to subtract the heat capacity of the addenda (chip + a few µg of Apiezon grease used to glue the sample onto the back of the chip), the empty chip was measured prior to the sample measurements. A precise in situ calibration and corrections of the thermometers in magnetic field were included in the data treatment. This technique enabled us to obtain absolute values of the specific heat of minute single crystals with an absolute accuracy better than ∼ 95 % as deduced from measurements on ultra pure copper (for further details, see [4]).

Sample Bi2201#2 has also been measured in pulsed magnetic fields up to 39 T and down to 0.6 K by the heat pulse method described in reference [START_REF] Kohama | Proc. Natl. Acad. Sci. USA[END_REF]. The single crystal was mounted on the sample platform with a small amount of Apiezon grease and the total heat capacity was estimated by C p = Q/∆T , where Q is the applied heat and ∆T the resultant temperature change. The addenda contribution originating from the sample platform and Apiezon grease was estimated by a separate experiment. The thermometer was calibrated by isothermal measurements of the magnetoresistance and the accuracy of C p was estimated to ∼ 93% below 2 K and ∼ 90% between 2 and 4 K, as deduced from measurements on a reference sample of polycristaline Ge [START_REF] Kohama | Proc. Natl. Acad. Sci. USA[END_REF].

III. RESULTS AND DISCUSSION

A. La2-xSrxCuO4

Efforts to measure the normal state specific heat at low temperature in LSCO have been hindered by the very large upper critical field values exceeding 60 T around optimal doping (for 0.14 ≤ p ≤ 0.21 [START_REF] Wang | [END_REF], see also [13] and references therein). This implies that measurements in the normal state have been limited to highly underdoped or overdoped samples in which superconductivity is absent [5,6,18] or weak enough to be easily suppressed Table I. Sample name, critical temperature Tc, chemical substitution rates (x and y), estimated T = 0 upper critical field ( * from [13,[START_REF] Wang | [END_REF]), mass, and B coefficient in the Bln(T0/T ) contribution to the specific heat of the Bi2+ySr2-x-yLaxCuO 6+δ and La2-xSrxCuO4 single crystals. Bi2201 samples #1 and #4 are the same as OD10K and OD18K in [11], respectively. Bi2201 samples #2 and #3 are the same as those for which the NMR Knight shift was reported in [15], yielding estimates of the pseudogap temperature T as plotted in Fig. 6. Last line : data from [4] by moderate magnetic fields [6,[START_REF] Wang | [END_REF]. However, ARPES measurements indicated that the pseudogap critical point 0.17 ≤ p ≤ 0.22 [19,20]. Accordingly, resistivity measurements suggest that p = 0.19 ± 0.02 [21][22][23], i.e. lying in the doping range for which the normal state is unattainable By applying magnetic fields up to 35 T, we were able to determine the specific heat in the normal state for p values as close as possible to p . Indeed, as shown in Fig. 1, the specific heat increases with field in the mixed state and saturates at high fields, clearly indicating that the applied fields are large enough to suppress superconductivity (or are slightly below H c2 (0) for p ∼ 0.22 and 0.145). Note that a H 2 field dependence is observed at the lowest temperatures for H ≥ H c2 (0), indicating the presence of a hyperfine Schottky contribution :

C hyp = AH 2 /T 2 with A ∼ 4 ± 1 × 10 -3 mJmol -1 K -2 (
see supplemental materials and Fig. 3 below). This hyperfine term is unfortunately hindering any unambiguous determination of the electronic contribution at very low temperature and we have here limited our lowest temperature to ∼ 2 K for the analysis of C e /T as C hyp /T becomes negligible above this temperature (see Fig. 4 in supplemental materials for the relative contribution of each term).

Our measurements then clearly indicate that the normal state electronic contribution to the specific heat displays a maximum upon doping (see Fig. 2 (a)), similar to the one observed in Nd and Eu substituted LSCO [4] (Fig. 2 (b)). As shown, very similar C e /T values are obtained for p ≥ 0.25, but clear differences are visible for lower dopings. Indeed, the position of the peak is shifted towards lower dopings in LSCO compared to its Nd and Eu substituted counterparts, due to the lower p value.

However, the maximum is significantly broader in LSCO, in agreement with the p dependence of the electronic specific heat previously obtained in Zn substituted samples [12] (see Fig. 2 (c)). Even though their exists some uncertainty on the determination of p, it is worth noting that those large C e /T values have been observed in samples with upper critical fields ranging from H c2 ∼ 14 T to H c2 ≥ 34 T (see Fig. 1) unambiguously confirming that C e /T remains large over an extended doping range (from p ≈ 0.20 to p ≈ 0.26).

Finally, note that attempts to determine the electronic contribution from high temperature specific heat measurements (i.e. for T > T c ) led to unconclusive results for p = 0.20. Those determinations were based on a difficult phonon subtraction procedure and reported values vary from 7 [24] to 14 mJmol -1 K -2 [25]. The latter value seems to be consistent with our work (see Fig. 2), but C e /T actually decreases with T , hindering any direct comparison between those high T values and those obtained here for T → 0.

The change of topology of the Fermi surface at p vHs p is expected to give rise to a strong (diverging) van Hove singularity in a purely 2D system [26] (see black line in Fig. 2 (d)). In fact, the amplitude of this singularity is strongly reduced both by the dimensionality and the presence of disorder. The dark-blue line in Fig. 2 (d) is a calculation for an anisotropic 3D system with an transverse hopping rate t z = 0.07t, t being the nearestneighbour hopping parameter in the tight binding Hamiltonian. The light-blue and purple lines are calculations for scattering rates τ equal to : /τ = 0.04t and 0.28t, respectively [26] (the latter value is consistent with the measured residual resistivity ρ 0 ∼ 30 µΩcm). We deduce from Fig. 2 (d) that the computations poorly reproduce (from [4]). (c) same data as in panel (a) together with data previously obtained in samples in which superconductivity was destroyed by Zn impurities [12] ((red) closed circles). (d) same data as in panel (a) together with calculations for a van Hove singularity at pvHs = 0.2 (from [26]) (solid lines) for the indicated geometries (2D/3D) and scattering rates ( /τ , see [26] for details). As shown, the dependence obtained for any van Hove singularity only very poorly reproduces the experimental data.

the experimental data. Moreover, as disorder is very different in pristine LSCO and Zn substituted samples, one should have observed a much reduced peak in the latter. This is not the case, hence strongly suggesting that the observed peak is not related to this van Hove singularity. Finally, it is interesting to compare our data on LSCO at p = 0.04 with recent quantum oscillation measurements in the 5-layer cuprate Ba 2 Ca 5 Cu 10 (F,O) 2 [27]. The frequency of the oscillations coming from the innermost layer (labelled IP0), in which there is long-range antiferromagnetic order, is F = 147 T. ARPES measurements show that the Fermi surface in this metal with antiferromagnetic order consists of four small closed hole pockets at nodal locations in the Brillouin zone, centered approximately at (±π/2, ±π/2) [27]. The 2D carrier density n contained by each hole pocket is given by n = F/Φ 0 , where Φ 0 = h/2e. The hole concen- The hyperfine (AH 2 /T 3 ) contribution becomes fully negligible above 2 -3 K (see for instance p = 0.12). The data at p = 0.29 and p = 0.33 (open symbols) are from [6] and [5], respectively (see also supplemental materials for other compositions).

tration (doping) p should then be given by p = 2n, given that there are two hole pockets per magnetic Brillouin zone, which yields p 0.04. The effective mass extracted from the quantum oscillations is m * ∼ 0.7m e , where m e is the bare electron mass. This corresponds to a specific heat Sommerfeld coefficient γ = 2 × 1.43 × m * ∼ 2 mJmol -1 K -2 mol-Cu, where the factor 2 comes from having two hole pockets in the Brillouin zone. This expected value of γ is not far from the value of 3 mJmol -1 K -2 mol-Cu reported here for LSCO at p = 0.04 (Fig. 1 and2). Given that the carrier density estimated from the Hall number in LSCO at p = 0.04 is indeed n = n H 0.04 [28], the Fermi surface of LSCO at p = 0.04 should be the same as that observed in the inner plane of Ba 2 Ca 5 Cu 10 (F,O) 2 , i.e. four nodal hole pock-ets containing p carriers, even though the commensurate antiferromagnetic phase in LSCO ends at p 0.02. This points to a similar Fermi surface in the pseudogap phase, beyond p 0.02, and in the antiferromagnetic phase, as suggested by Hall studies in various cuprates [8,9,11] and associated calculations [29][30][31][32].

A second indication for the presence of a QCP is the occurence of a ln(1/T ) contribution in the temperature dependence of the specific heat [4]. As shown in Fig. 3, the data can be well described by the standard C/T = γ 0 + βT 2 law above ∼ 2 K for dopings far from the critical doping p , i.e. for p ≤ 0.145 and p ≥ 0.29 (a small δT 4 correction has to be introduced above ∼ 5 K). At very low temperature, a clear deviation due to the hyperfine contribution is visible, in good agreement with the H 2 dependence observed in C/T (H) (see Fig. 2 in the supplemental materials). In contrast with data at p = 0.12, the upturn at low temperature cannot be described by the hyperfine contribution alone for dopings close to p (i.e. for p = 0.22 -0.25), indicating an extra contribution which extends well above 2 K. Note that a very similar contribution is also visible in the data previously obtained by Wang al. [6] for p ∼ 0.26 (see Fig. 1 in supplemental materials). Such an upturn has also been observed in Zn-LSCO (below ∼ 3.5 K, for x ∼ 0.2) and has been attributed to local magnetic moments induced by Zn doping [12], but this interpretation can not hold in our pristine LSCO crystals.

In this doping range, the data can then be very well fitted by introducing an extra ln(1/T )) contribution to the specific heat, C/T = AH 2 /T 3 + βT 2 + δT 4 + B ln(T 0 /T ) where the constant Bln(T 0 ) term contains both the band structure Sommerfeld contribution and a cutoff temperature above which the contribution of the (quantum) fluctuations vanishes (see solid lines in Fig. 3). To clearly highlight this logarithmic contribution, it is displayed in Fig. 3 (a) as a shaded area under the curve for p = 0.25. On the contrary, a very good fit to the data can be obtained with B = 0 for p = 0.12 (see dashed line in Fig. 3 (b)).

B. Bi2+ySr2-x-yLaxCuO 6+δ

We performed similar measurements on four Bi2201 crystals (see Tab. 1). In Bi2201, the hyperfine contribution to the specific heat we observe is much weaker than in LSCO (A = 8 ± 1 × 10 -4 mJmol -1 K -2 ) and the normal state could be reached in all measured samples (see Table 1 and Fig. 3 in the supplemental materials for the upper critical field H c2 (0) values). As an example, in the normal state of sample Bi2201#1, at a field just above H c2 (0) ∼ 15 T, the hyperfine term is about 20 times smaller than in LSCO. The contribution of the ln(1/T ) term can then be straightforwardly identified in the temperature dependence of C/T and is highlighted by a shaded area in Fig. 4 (a) (see Fig. 4 in supplemental materials for the relative contribution of each term). Complementary measurements have also been performed in Bi2201#2 in pulsed magnetic fields up to ∼ 39 T (see Fig. 4 (b)). Although slightly larger, the obtained C/T values are in reasonable agreement with the one obtained in DC field. They confirm the presence of a clear upturn at low temperature and the absence of any significant contribution from the hyperfine term (i.e. no field dependence) up to 39 T on the whole temperature range. In all samples, C/T can then be very well fitted by a AH 2 /T 3 + βT 2 + B ln(T 0 /T ) law (solid lines in Fig. 4). In samples Bi2201#1-3, the amplitude of the logarithmic contribution is identical within error bars (see Table 1) and this contribution is only significantly smaller in Bi2201#4, which has the highest T c and hence a doping content further away from p (see shaded area in Fig. 4 (d)).

As pointed out by Lohneysen in the context of heavy fermion systems [33], the universality of the critical exponents of a given universality class is expected to trans-late into a constant B value for all compounds belonging to this universality class. It is then important to note that we obtained very similar B coefficients (∼ 2 -3 mJmol -1 K -2 close to p ) in LSCO, Nd/Eu-LSCO and Bi2201 (see table 1), strongly indicating that they all belong to the same universality class. Moreover, following Varma [34], B is expected to be on the order of the band structure Sommerfeld coefficient γ 0 and our B value is hence quantitatively consistent with a γ 0 value being on the order of 5 mJmol -1 K -2 as measured far from p (the coupling constant g = B/γ 0 ∼ 0.5 is then in agreement with transport and photoemission measurements, see [34] for a detailed discussion).

On the other hand, Bln(T 0 ) varies from ∼ 11 -12 mJmol -1 K -2 in Bi2201 to ∼ 16 -19 mJmol -1 K -2 in lanthanum based cuprates (see Fig. 5). This increase in T 0 is consistent with Seebeck coefficient measurements [11,35] showing that, at low temperature, S(T ) is significantly larger in Nd-LSCO than in Bi2201. S(T ) also displays a ln(1/T ) temperature dependence but this contribution is then flattening off above ∼ 30 K in Bi2201 whereas it extends up to ∼ 100 K in Nd-LSCO. Finally, note that we observe a small increase of T 0 with field in Bi2201#3 (see Fig. 4 (c) and Fig. 5). Although small, this field induced change can not be attributed to the Schottky contribution (see Fig. 5 in supplemental materials) and still has to be understood (see also concluding remarks below).

Finally let us turn to the doping dependence of C e /T in Bi2201. It is difficult to determine an unambiguous doping content p in Bi 2+y Sr 2-x-y La x CuO 6+δ due to the various doping routes (x, y and δ) and the phase diagram has hence been plotted as a function of the critical temperature T c in Fig. 6. As shown, the electronic specific heat is on the order of 8 ± 1 mJmol -1 K -2 at 3 K but rapidly increases with decreasing T , reaching ∼ 13 ± 1 mJmol -1 K -2 at 0.65 K in Bi2201#1-3. As shown, C e /T (at 0.65 K) then decreases in Bi2201#4. Bi2201#1-3 clearly lie close to the onset of the pseudogap phase as deduced from ARPES [36] and NMR [37] studies, emphasising the enhancement of C e /T (at low T ) in the vicinity of this point. Note that a linear temperature dependence of the resistivity has been reported recently in Bi2201#1 (labelled OD10K) [11] but this linearity does not persist down to T = 0 K, indicating that the doping content is slightly below p for this T c value.

It is interesting to further compare our specific heat data on Bi2201 with recent transport measurements performed on same samples (sample labelled OD10K and OD18K in ref. [11] are the same as Bi2201#1 and Bi2201#4, respectively). In particular, the Hall effect measurements yield a large difference in Hall number between the two, namely n H ∼ 1.4 for Bi2201#1 and n H ∼ 0.75 for Bi2201#4, interpreted as a sharp, nearly two-fold, drop in carrier density with decreasing p [11]. A similar decrease has been reported by Putzke et al.

[10] and a very similar drop was observed previously in YBCO [8] and Nd-LSCO [9]. This drop has been identified as a key signature of the pseudogap phase that reveals a transformation of the Fermi surface across p , consistent with a change from a large surface containing 1+p holes to small Fermi pockets containing p holes [29,38]. Note that for the same Bi2201 samples, the electronic specific heat at T = 3 K changes very little from one doping to the other (Fig. 6). This is not necessarily incompatible with a transformation of the Fermi surface. Indeed, calculations show that when a metal undergoes a transition into a phase of antiferromagnetic order (at T 0) [31], the specific heat is barely affected initially whereas the Hall number exhibits a rapid drop.

Another property of Bi2201 that was measured in the field-induced normal state at T → 0 is the NMR Knight shift [37], which reflects the electronic spin susceptibility, typically assumed to be proportional to the electronic density of states. In that sense, it should be closely related to the electronic specific heat reported here. What we observe is that the two quantities do not show the same degree of change below p . Indeed, while the electronic spin susceptibility at T 2-3 K drops by a factor of about 20 % in going from T c = 10 K to T c = 18 K [37], the electronic specific heat does not change (Fig. 6). This suggests that the two measurements, performed in the very same conditions of temperature and field, on the same Bi2201 samples, do not reflect the same underlying physical quantity, i.e. they do not both reflect the same density of states. Understanding this difference, revealed here for the first time in a cuprate material, should provide new insight into the nature of the pseudogap phase.

IV. CONCLUDING REMARKS

In summary, we have shown that the electronic contribution to the normal state specific heat displays a ln(1/T ) temperature dependence associated to a strong increase of C e /T for T → 0, close to the critical doping p that marks the onset of the pseudogap phase in both LSCO and Bi2201, as previously observed in Nd/Eu-LSCO [4]. These features seem therefore generic in cuprates and are classical signatures of the existence of a QCP, whose nature has not yet been identified. In LSCO, the ln(1/T ) term is observed up to 0.26 i.e. well above p = 0.19 ± 0.02. This extended doping range could be reminiscent of the anomalous form of criticality previously pointed out by Cooper et al. in transport measurements [23], for which a a linear term in the temperature dependence of the resistivity is observed from p ∼ 0.18 up to p ∼ 0.3 [23].

Moreover, we observed that the criticality is reinforced by the magnetic field in Bi2201 (the large hyperfine contribution and H c2 values are hindering a similar study in LSCO) as the electronic contribution to the specific heat for Bi2201 samples #1 (black), #2 (green), #3 (blue) and #4 (red) (same color code as in Fig. 4), together with the evolution of the pseudogap phase onset temperature T , as deduced from ARPES (diamonds from [36]) and NMR (circles from [15,37]) measurements. Closed symbols correspond to 18 T measurements and open symbols to 32 T measurements, suggesting a (small) increase of the electronic contribution with field in the normal state (see Fig. 5).

slightly increases with field in the normal state. This puzzling behaviour is in contrast to the one usually observed in heavy fermion systems in the vicinity of a magnetic QCP [39]. Indeed, in this case the quantum fluctuations are suppressed by the application of magnetic field and the ln(1/T ) contribution saturates at low T for large H values. Recent NMR and sound velocity measurements [13] indicated that the short-range magnetism is reinforced under high magnetic fields in LSCO. The interplay between this magnetic state and the signatures of quantum criticality observed here still has to be clarified, but it is worth noting that this magnetic phase vanishes at a critical doping which extrapolates to p for large magnetic field leading to a possible field dependence of the QCP (see also [40]). However, no indication for the presence of such a magnetic state in the vicinity of the onset of the pseudogap phase has been observed in Bi2201.

Finally, calculations [41] for the two-dimensional Hubbard model in the doped Mott insulator regime have proposed an alternative interpretation of these thermodynamic anomalies without invoking broken symmetries. A sharp cross-over in the specific heat can occur when crossing the Widom line emanating from a finite temperature critical endpoint of a first-order transition between a pseudogap phase and a "standard" metal with Fermi liquid correlations [42].

The work in Grenoble was supported by the Laboratoire d'excellence LANEF (ANR-10-LABX-51-01) and was performed at the LNCMI, a member of the European Magnetic Field Laboratory (EMFL). Work at the LNCMI was supported by the French Agence Nationale de la Recherche (ANR) (contract ANR-19-CE30-0019). L.T. acknowledges support from the Canadian Institute for Advanced Research (CIFAR) and funding from the Natural Sciences and Engineering Research Council of Canada (NSERC; PIN:123817), the Fonds de recherche du Quebec Nature et Technologies (FRQNT), the Canada Foundation for Innovation (CFI), and a Canada Research Chair. This research was undertaken thanks in part to funding from the Canada First Research Excellence Fund. Part of this work was funded by the Gordon and Betty Moore Foundation's EPiQS Initiative (Grant GBMF5306 to L.T). J.C. acknowledge support from the Swiss National Science Foundation).

Figure 1 .

 1 Figure 1. Magnetic field dependence of the specific heat at T ∼ 2.1 K in La2-xSrxCuO4 (sample details are shown in Tab. 1), after subtraction of the phonon contribution C ph (see Fig.3). The small overshoot observed for p ∼ 0.24 is most probably reminiscent of the superconducting transition in presence of strong fluctuations. The shaded (colored) boxes indicate the locus of the previously estimated Hc2(0) values[13,[START_REF] Wang | [END_REF]. Open circles are low field data extracted from[6]. As seen, (C -C ph )/T increases with p, reaching ∼ 15-16 mJmol -1 K -2 in the normal state for p ∼ 0.22-0.25 at T ∼ 2 K (see also Fig.2).

Figure 2 .

 2 Figure 2. (a) Electronic contribution to the specific heat Ce/T (at 2 K) as a function of the doping content p in La2-xSrxCuO4 (solid squares, see Tab. 1 for sample details)together with data previously obtained in samples of lower Hc2 values[5,6,18] (crossed open squares). The same guide to the eyes (thin lines) is used in all panels. (b) same data as panel (a) together with data previously obtained in Nd-LSCO (black circles) as well as Eu-LSCO (black squares) crystals (from[4]). (c) same data as in panel (a) together with data previously obtained in samples in which superconductivity was destroyed by Zn impurities[12] ((red) closed circles). (d) same data as in panel (a) together with calculations for a van Hove singularity at pvHs = 0.2 (from[26]) (solid lines) for the indicated geometries (2D/3D) and scattering rates ( /τ , see[26] for details). As shown, the dependence obtained for any van Hove singularity only very poorly reproduces the experimental data.

Figure 3 .

 3 Figure 3. Temperature dependence of the specific heat obtained in La2-xSrxCuO4 for p ≥ 0.25 (panel a) and p ≤ 0.22 (panel b) (see Tab. 1 for sample details).For p = 0.22 and 0.25, the data can be very well described by a C/T = AH 2 /T 3 + βT 2 + δT 4 + B ln(T0/T ) law ((black) solid lines). The (black) dashed line in panel (a) corresponds to the dependence that would be obtained for B = 0 (p = 0.25) and the shaded area then highlights the contribution of the Bln(T0/T ) term. On the other hand a very good with to the data can be obtained with B = 0 for p = 0.12 ((black) dashed line in panel (b)). The standard C/T = γ0 +βT 2 dependence is indicated by the dotted lines (panel b). The hyperfine (AH 2 /T 3 ) contribution becomes fully negligible above 2 -3 K (see for instance p = 0.12). The data at p = 0.29 and p = 0.33 (open symbols) are from[6] and[5], respectively (see also supplemental materials for other compositions).

Figure 4 .

 4 Figure 4. Temperature dependence of the specific heat of our Bi2+ySr2-x-yLaxCuO 6+δ single crystals (see Tab. 1 for sample details). Solid lines are fit to the data assuming that C/T = AH 2 /T 3 + βT 2 + B ln(T0/T ) and the dashed line correspond to the dependence that would be obtained for B = 0. The contribution of the B ln(T0/T ) term is highlighted by the grey shaded area. As shown panel (d), the contribution of the ln(1/T ) term is much weaker in Bi2201#4 of larger Tc value. Complementary pulsed field data for fields ranging from 12 T to 39 T (Bi2201#2) are also displayed in panel (b). Measurements of Bi2201 at both 18 T and 32 T are displayed in panel (c), indicating a field dependence of C/T in the normal state, well above 1 K (see also Fig. 6 and text for details).

Figure 5 .Figure 6 .

 56 Figure 5. Temperature dependence of the electronic contribution to the specific heat Ce/T = C/T -AH 2 /T 3 -C ph /T of Bi2201#3, compared to the dependence previously obtained in Eu-LSCO (from [4]), clearly displaying the logarithmic decrease (Ce/T = B ln(T0/T )), with a field dependent T0(H).

  .

	name	Tc (K)	x	y	Hc2(0) (T)	mass (mg)	B (mJmol -1 K -2 )
	Bi2201#1	∼ 11.5	0	0.05	∼ 15	0.42	2.8 ± 0.5
	Bi2201#2	∼ 10	0.04	0	∼ 15	1.20	3.2 ± 0.5
	Bi2201#3	∼ 13.5	0.08	0	∼ 17	0.37	2.5 ± 0.5
	Bi2201#4	∼ 18	0.20	0	∼ 22 (at 2.1 K)	0.39	< 1
	LSCO#1	0	0	-	0	0.60	0
	LSCO#2	0	0.04	-	0	0.95	0
	LSCO#3	∼ 20	0.12	-	∼ 19	0.75	0
	LSCO#4	∼ 25	0.145	-	40 ± 5 *	1.50	0
	LSCO#5	∼ 25	0.22	-	37 ± 4 *	0.94	2.0 ± 0.3
	LSCO#6	∼ 16	0.24	-	∼ 23 (at 2.1 K)	1.50	2.2 ± 0.3
	LSCO#7	∼ 15	0.25	-	∼ 14 (at 2.1 K)	0.85	2.1 ± 0.3
	Eu-LSCO	11	p = 0.24		11	-	2.5 ± 0.5


Juliette Grosset

Stephane Mazzotti

Philippe Vernant

Juliette Grosset

Julie&e Grosset
email: juliece.grosset@umontpellier.fr

Stéphane Mazzo7

Estimation of the present-day deformation of the Western Alps

Introduction datasets Strain rates

The Western Alps is one of the most seismically ac3ve regions in western Europe with a large density of earthquakes and some historical damaging earthquakes (example: Bâle earthquake, 1356). The seismic strain computed from focal mechanisms shows a E-W to NW-SE extension in the French Alps, and a N-S extension in Southern Switzerland [START_REF] Delacou | Present-day geodynamics in the bend of the western and central Alps as constrained by earthquake analysis[END_REF].

Reference

The identification of robust strain rate patterns in the Western Alps is made difficult by the low deformation rate in the region, hard to distinguish from the noise [START_REF] Masson | Extrac3ng small deforma3on beyond individual sta3on precision from dense Global Naviga3on Satellite System (GNSS) networks in France and western Europe[END_REF][START_REF] Walpersdorf | Does Long-Term GPS in the Western Alps Finally Confirm Earthquake Mechanisms?[END_REF] the precise characterization of present-day strain rate, and its spatial variability, is a crucial point to understand the origin of deformation.

The main goal of this study is to identify a robust strain rate model by the re-evaluation of the GNSS horizontal velocities with recent data and the comparison of different datasets to identify common signals. We bring a comparison analysis on three velocity fields computed from three different networks.

GM NGL ST

Ø Extension and strike-slip strain rates in the inner Alps, and shortening rates in the foreland.

Ø A N-S extension rate area is present on the three maps on the France-Switzerland-Italy border, from about 1.7 to 2.3 × 10-9 yr-1

Ø The extension rate is associated in the French Alps with strike-slip (GM, NGL)

Ø Very low deformation rates in northern Switzerland.

First-order synthetic strain rate map interpreted from the comparison of the 3 GNSS datasets. Demarcated area and arrows amplitude are extracted from the 3 strain rate map.

Geode3cal strain rate vs. Seismic strain ? What are the processes of deforma3on in the Western Alps ?


Joar Øveraas Halvorsen

MD Florian Naudet

PhD Ioana A Cristea

cand.psychol Joar Øveraas Halvorsen
email: joar.halvorsen@ntnu.no

Matters Arising: Challenges with benchmarking of MDMA-assisted psychotherapy 1

Main:

To represent a treatment breakthrough, MDMA-assisted psychotherapy for posttraumatic stress disorder should be evaluated against first-line psychological interventions or for pre-specified patient subgroups that do not improve after such interventions.

Mitchell et al. [START_REF] Mitchell | MDMA-assisted therapy for severe PTSD: a randomized, double-blind, placebo-controlled phase 3 study[END_REF] recently reported short-term results from a phase 3 trial of MDMAassisted psychotherapy for posttraumatic stress disorder (PTSD), concluding that "[c]ompared with current first-line pharmacological and behavioral therapies, MDMA-assisted therapy has the potential to dramatically transform treatment for PTSD and should be expeditiously evaluated for clinical use". PTSD is a chronic and disabling condition and identifying novel beneficial therapies is timely and important. New treatments could prove useful by being more effective for symptoms or other patient-relevant outcomes (e.g., functioning, quality of life), more cost-effective, or more acceptable to patients (e.g., due to less side-effects). Any of these advantages could apply either to patients overall or to circumscribed subgroups, particularly when these include individuals for whom existent therapies do not work well. However, evaluating new treatments on these parameters necessitates comparing them to interventions currently recommended as "first-line". Benchmarking against the best currently available treatments is fundamental particularly for labeling a new treatment as a "breakthrough", a term with powerful connotations for patients, clinicians and regulators. For PTSD, the current best available treatments are represented by psychological interventions, currently considered as first line treatments for the disorder by most major clinical guidelines such as the American Psychological Association 2 and the National Institute for Health and Care Excellence (NICE) 3 . These guidelines recommend a number of trauma-focused psychological treatments (TFPs), including prolonged exposure therapy (PE), cognitive processing therapy (CPT), eye movement desensitization and reprocessing (EMDR) and trauma-focused cognitive behavioral therapy (TF-CBT).

In terms of comparative effectiveness, Mitchell et al. reported a reduction in PTSD symptoms (standardized mean difference/SMD) of 0.91 (95% CI 0.44-1.37), which they contrast to the modest effects of some pharmacological treatments, like sertraline (SMD=0.51, 95% CI 0.38-0.64) and paroxetine (SMD=0.36, 95% CI 0.28-0.49) [START_REF] Lee | Psychotherapy versus pharmacotherapy for posttraumatic stress disorder: Systemic review and meta-analyses to determine first-line treatments[END_REF] . However, first-line interventions like TFPs are significantly more effective than antidepressants, with SMDs versus control of 0.83 (95% CI 0.69-0.97) [START_REF] Lee | Psychotherapy versus pharmacotherapy for posttraumatic stress disorder: Systemic review and meta-analyses to determine first-line treatments[END_REF] . A recent network meta-analysis [START_REF] Mavranezouli | Psychological treatments for post-traumatic stress disorder in adults: a network meta-analysis[END_REF] showed even greater effects on PTSD symptoms for several psychological treatments compared to waitlist, including EMDR (SMD=2.07, 95% CrI 1.44-2.70) and TF-CBT (SMD=1.46, 95% CrI 1.05-1.87). Similarly, in another meta-analysis [START_REF] Karatzias | Psychological interventions for ICD-11 complex PTSD symptoms: systematic review and meta-analysis[END_REF] , psychological interventions like CBT (SMD= 0.90; 95% CI 0.68-1.11), exposure therapy alone (SMD=1.05; 95% CI 0.58-1.52) and EMDR (SMD=1.26; 95% CI 0.512.01) were superior to usual care in patients with complex PTSD. Thus, these psychological interventions, which attain similar or higher symptom reduction compared to MDMA-assisted psychotherapy, would represent an appropriate comparator for judging comparative effectiveness.

Examination of another clinically relevant outcome, remission or loss of diagnosis, points to a similar picture. Again, for several first-line psychological treatments, rates are higher than the 33% post-treatment remission reported by Mitchell et al. For example, Ehlers et al. [START_REF] Ehlers | A Randomized Controlled Trial of 7-Day Intensive and Standard Weekly Cognitive Therapy for PTSD and Emotion-Focused Supportive Therapy[END_REF] reported post-treatment remission rates ranging from approximately 46% to over 70%, depending on mode of assessment, for two versions of cognitive therapy. A meta-analysis 8 of CBT for PTSD reported a mean remission rate of around 53% (95% CI 45%-61%). Furthermore, Resick et al. [START_REF] Resick | Long-term outcomes of cognitive-behavioral treatments for posttraumatic stress disorder among female rape survivors[END_REF] demonstrated a remarkable maintenance of effects over an extensive long-term follow-up for both CPT and PE, with only 22.2% and 17.5% respectively of the intent-to-treat sample of female rape survivors still qualifying for a diagnosis.

Once a novel treatment is proven effective, and particularly if deemed a breakthrough, large-scale dissemination is to be expected. Therefore, two additional aspects to consider are adverse effects (AE) and cost-effectiveness. For the first, serious adverse effects associated with MDMA use reported in Mitchell et al. were rare. However, although rare events are difficult to evaluate reliably in phase 3 trials, due to limited sample sizes and lack of long-term follow-up, they can become noticeable when a treatment is widely implemented. Given that the abuse potential and adverse effects of MDMA, even with limited use, are substantial 10 , regulators should require comprehensive evidence on safety and rely on more evidence than a single small study to define an adequate post-approval risk management plan.

Regarding the second aspect, though cost-effectiveness of MDMA-assisted psychotherapy was not yet formally evaluated, it is worth underscoring that the amount of therapy involved is greater than for several first-line psychological interventions. The psychotherapy component in the trial consisted of three preparatory 90 minutes sessions, three 8-hours sessions of delivering MDMA-assisted psychotherapy, each followed by three 90 minutes integration sessions. Overall, the psychotherapy exposure was equivalent to 28 90 minutes sessions or 42 60-minutes sessions. In addition, the presences of two therapists were required in all sessions. Conversely, existing first-line psychological treatments for PTSD, discussed previously, usually consist of 8 to 16 sessions of 60 to 90 minutes duration with an individual therapist [START_REF]Clinical practice Guideline for the Treatment of Posttraumatic Stress Disorder (PTSD) in Adults[END_REF]3 or up to 20 hours of therapy [START_REF] Ehlers | A Randomized Controlled Trial of 7-Day Intensive and Standard Weekly Cognitive Therapy for PTSD and Emotion-Focused Supportive Therapy[END_REF] , amounting to half or less than required by MDMA-assisted therapy.

Moving forward, a judgement as to whether MDMA-assisted psychotherapy for PTSD represents a true therapeutic breakthrough requires a phase 3 program that incorporates large pragmatic studies with adequate comparators, like trauma-focused psychological therapies.

Alternatively, the therapy could be tested in rigorously pre-specified subgroups of patients that did not respond to adequate courses of first-line treatments, like TFPs. Given the chronic nature of PTSD and its pervasive and durable impact on patients' lives, trials should also assess patient relevant outcomes beside symptoms, like quality of life, and include mid-and long-term follow-ups. Finally, a thorough investigation of any potential safety issues should be carried out on large samples and at over longer timeframes to ensure a reliable evaluation of the balance of benefits and risks.

Author contributions:

JØH and IAC conceptualized the main arguments, and JØH wrote the first draft. All authors contributed substantially to the revisions of the manuscript.

Ethics statement:

The authors declare no financial or non-financial conflict of interests.


Christine Joblin

Carbon macromolecules in the cycle of interstellar matter : observations and laboratory experiments

HAL is

Astrophysical context

The Aromatic Infrared emission Bands (AIBs) between 3.3 and 12.7 µm are well observed in UV-irradiated regions. The identity of their carriers is not precisely established but polycylic aromatic hydrocarbons (PAHs) remain the best candidates. Energetic budget calculations have led to estimates of 10 to 20% of total carbon contained in these species, making them an important component of interstellar matter. Trying to match the AIBs with the spectra of laboratory samples does not appear to be a successful method to identify interstellar PAHs. One has to think where these species come from and how they evolve due to environmental conditions (UV irradiation, gas and dust interactions). Several topics can be addressed: -i-PAHs contain hydrogen at their periphery. Photodissociation can affect the hydrogenation coverage. Is this a mechanism to produce H 2 ? -ii-Is the fragmentation of the carbon skeleton of PAHs a reservoir to produce small gas-phase species? -iii-What is the connection between PAHs and larger structures (the VSGs of Désert et al. 1990)? -iv-PAHs can play an important role in chemistry. They offer a large surface and can contain some very active dangling bonds. As the AIBs are the only evidence for the presence of PAHs, infrared spectroscopy is the obvious diagnostic to answer these questions. Since dependence on physical conditions (UV field and gas density) is searched for, extended photodissociation regions (PDRs) are good objects to be observed providing additional imaging capabilities. Examples are given in the next section. The astronomical media considered are characterized by low densities, typically less than 10 6 cm -3 and cold temperatures, a few tens of K for PAHs between the absorption of two UV photons. PIRENEA is an experimental set-up that was developed according to these specifications (high isolation and low temperatures) by combining the trapping capabilities of an ion cyclotron resonance (ICR) cell with cryogenic shielding. Some recent results obtained with PIRENEA are presented in section 3.

Questions raised by observations

Items (i) and (iv) invoke an evaluation of the role of PAHs in the formation of H 2 . The spatial distribution of the emission in the AIBs has been compared by several authors with that in the H 2 ro-vibrational lines in the near-IR (Brooks et al. 2000[START_REF] Habart | [END_REF], Joblin et al. 2003a). Comparison with the predictions of PDR models underlies the difficulty to account for the H 2 emission observed in regions where the AIBs are bright. It has therefore been proposed that PAHs in these regions probably contribute to the formation of H 2 [START_REF] Habart | [END_REF]. The various mechanisms that can take place have been reviewed by the authors: reaction of H atoms (free or physisorbed) with chemisorbed H (Cazaux & Tielens 2002) or photodissociation by UV photons [START_REF] Joblin | Molecular Hydrogen in Space[END_REF]. The question of the coupling of PAHs with small hydrocarbons (item -ii-) has recently gained some interest thanks to the detection with radiotelescopes of species such as C 2 H, c-C 3 H 2 , C 4 H and C 6 H in PDRs (Fossé 2003). The observed abundances cannot be accounted for by PDR models [START_REF] Gerin | Chemistry as a Diagnostic of Star Formation[END_REF]. Could the photodestruction of PAHs replenish the gas-phase in small hydrocarbons or carbon clusters? Another major question is that of the formation of PAHs. Where do these molecules so abundant in PDRs come from? The analysis of data on reflection nebulae provided by the camera CAM on board the European Infrared Space Observatory brings us some new clues. Bright AIBs are observed at the cloud interfaces but an additional continuum appears to rise when going inside the cloud (Abergel et al. 2002). Using a singular value decomposition analysis, a full spectrum was extracted for the additional emitting species (Boissel et al. 2001, Rapacioli et al. these proceedings). There is a continuum rising towards longer wavelengths but also associated bands which are generally broader than the AIBs but keep some similarities. Furthermore, the decrease in the emission of this component at the border of the cloud is found to correlate with the increase in the emission of the AIBs. This led the authors to propose that the species carrying the continuum emission are VSGs formed of PAH clusters which evaporate at the border of clouds and release free PAHs (cf. Rapacioli et al.). This sets item -iii-.

Laboratory approach

Questions raised by observations can be addressed in the laboratory. The PIRENEA set-up was specifically developed for studies related to large molecules such as PAHs or very small grains of astrophysical interest. A description of the experiment can be found in [START_REF] Joblin | SF2A, Scientific Highlights[END_REF]. At that time, the aim was to optimize the photophysical interface. It was shown that the photodissociation of the coronene PAH cation 12 C 24 H + 12 under continuous irradiation of a Xe lamp leads to preferential formation of even-mass species 12 C 24 H + 2n with n=[0,5]. It was not clear whether dehydrogenation was proceeding through H 2 formation or sequential loss of H, the odd-mass species being fragile intermediates easily destroyed under continuous irradiation. Using the capability of selective ejection provided by the ICR cell, more recent experiments confirm the latter explanation [START_REF] Joblin | [END_REF]). The C 2 H 2 channel was found to be negligible in the photodissociation of the compact PAH C 24 H + 12 , whereas it is known to be a major channel in the fragmentation of small species n≤14 [START_REF] Jochims | [END_REF]). For C 24 H + 12 , the destruction of the carbon skeleton occurs only after complete dehydrogenation. The fragmentation of C + 24 leads to a series of carbon clusters (Fig. 1). Since the experiment is performed under continuous radiation, some of them are nor primary daughter species but produced by further fragmentation. Calculations by Jones & Seifert (1997) provide the various isomeric forms of C 24 , the highest thermodynamic stability being attained by the coronene skeleton, closed cage fullerene and ring structures including the bicyclic ring 14C-10C. The presence of the latter form in our experiment is strongly suggested by the strong peak of C + 14 (Fig. 1). C + 11 is then formed by loss of C 3 . Looking at the spectrum in Fig. 1, it also appears that there are other dissociation paths involving different isomers of C + 24 and a dissociation cascade producing C, C 2 and C 3 . Although a specific study would be necessary for each species, the global picture appears to be consistent with previous studies (cf. for instance Shelimov et al. 1994;Pozniak & Dunbar 1997). First results were also obtained on the larger molecule dicoronene C 48 H + 20 . In this case again, the photodissociation proceeds until full dehydrogenation followed by loss of C 2 by the pure carbon cluster C + 48 (Fig. 1b). This is the usual dissociation channel for fullerenes, the prototypical example being C 60 (cf. review by Lifshitz 2000). Results have just been obtained with PIRENEA on the reactivity of PAH cations (including dehydrogenated species) with various gases, O 2 and H 2 O in particular. Studies dedicated to PAH clusters are planned in the near future.

Figure 1 :

 1 Figure 1: Photodissociation of (a) 12 C 24 H + 12 and (b) C 48 H + 20 (includes 13 C isotopomers) under continuous visible UV irradiation: (a) 120 s of Xe lamp + OPO laser tuned at 480 nm and (b) 30 s of Xe lamp + OPO laser tuned at 440 nm.

Acknowledgments

M. Armengaud and P. Frabel are acknowledged for their technical involvment in the PIRENEA set-up, the Programme National de Physique et Chimie du Milieu Interstellaire and the Région Midi-Pyrénées for their financial support. Special thanks to M. Gerin and D. Teyssier for involving me in observations of small hydrocarbons in PDRs and to J. Cernicharo for encouraging the studies on carbon clusters.


Fedor Fomin

Archontia C Giannopoulou

Roman Rabinovich

Spyros Angelopoulos

Nancy Clarke

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Preface to the special issue on Graph Searching: Theory and Applications

The idea of this special issue of Theoretical Computer Science originates from the 9th International Workshop on GRAph Searching: Theory and Applications (GRASTA 2018) that took place in Berlin, Germany between the 24th and 27th of September, 2018. The issue contains results that were presented during GRASTA 2018 as well as external contributions on Graph Searching. GRASTA 2018 took place in Berlin, Germany between the 24 th and 27 th of September, 2018. The workshop was established in 2006 and since then it has traveled around the world, being held in Anogia, Crete, Greece (2006 and2017), Praia de Redonda, Ceara, Brazil (2008), Valtice chateau, Valtice, Czech Republic (2009), Dagstuhl, Germany (2011), Banff, Canada (2012), Cargèse, Corsica, France (2014), Montréal, Canada (2015) (joint with 5th workshop on Moving And Computing (MAC)) and Berlin, Germany (2018). It has become a regular meeting where results within the broad scope of Graph Searching are communicated and collaborations are promoted between a wide range of researchers from all over the world.

Graph Searching can be characterised as a game played by two opposing teams inside a network, modelled by a graph. The first team consists of mobile agents (called searchers or pursuers or cops) and the second team consists of escaping agents (called evaders or fugitives or robbers) residing and moving within the graph. Both the mobile and the escaping agents can be assigned attributes picking from a wide pool of options, including their respective relative speeds, sensor capabilities, visibility, etc. The graph searching games may then take vastly different forms. The pursuers win if they capture the evaders. Otherwise, the evaders win. The notion of capture itself admits several interpretations as well.

Many game variants have been studied in the literature and are either application driven or are inspired by foundational issues in Computer Science, Discrete Mathematics, and Artificial Intelligence. Moreover, they are related to diverse topics such as Robot Motion Planning, Database Theory, Logic, Distributed Computing, Models of Computation, and Network Security.

This Special Issue received 11 submissions in total. All submissions were reviewed according to the highest standards of Theoretical Computer Science, and among them 7 submissions were finally accepted for publication. The selected papers cover a wide range of topics such as: Capture Time Complexities of Graph Searching Games, Directed Graph Theory, Discrete Optimization, Game Theory, Graph Admissibility, Graph Decomposition Theorems, Online Algorithms and Scheduling.

A more detailed description of the accepted papers, in order of final acceptance date, is as follows.

• In the paper A search game on a hypergraph with booby traps, Thomas Lidbetter and Kyle Y. Lin consider the graph searching game that follows. A set of n boxes, located on the vertices of a hypergraph G, contain known but different rewards. A Searcher opens all the boxes in some hyperedge of G with the objective of collecting the maximum possible total reward. Some of the boxes, however, are booby trapped. If the Searcher opens a booby trapped box, the search ends and she loses all her collected rewards. The authors assume the number k of booby traps is known, and model the problem as a zero-sum game between the maximizing Searcher and a minimizing Hider, where the Hider chooses k boxes to booby trap and the Searcher opens all the boxes in some hyperedge. The payoff is the total reward collected by the Searcher. The authors solve the game when G is a 1-uniform hypergraph (the hyperedges are all singletons). When G is the complete hypergraph (containing all possible hyperedges), the game is solved in the cases where each box has the same reward, k = 1, and finally n = 4 and k = 2.

• In the paper Edge degeneracy: Algorithmic and structural results, the authors Stratis Limnios, Christophe Paul, Joanny Perret, and Dimitrios M. Thilikos consider a cops and robber game where the cops are blocking edges of a graph G, while the robber occupies its vertices. At each round of the game, the cops choose some set of edges to block and right after the robber is obliged to move to another vertex traversing at most s unblocked edges (s can be seen as the speed of the robber). Both parts have complete knowledge of the opponent's moves and the cops win when they occupy all edges incident to the robber's position. The authors introduce the capture cost on G against a robber of speed s. This defines a hierarchy of invariants, namely δ 1 e , δ 2 e , . . . , δ ∞ e , where δ ∞ e is an edge-analogue of the admissibility graph invariant. They prove that the problem asking whether δ s e (G) ≤ k, is polynomially solvable when s ∈ {1, 2, ∞} while, otherwise, it is NP-complete. Their main result is a structural theorem for graphs of bounded edge-admissibility.

• Sebastian Brandt, Klaus-Tycho Foerster, Jonathan Maurer, and Roger Wattenhofer in their paper

Online graph exploration on a restricted graph class: Optimal solutions for tadpole graphs study the problem of online graph exploration on undirected graphs, where a searcher has to visit every vertex and return to the origin. Once a new vertex is visited, the searcher learns of all neighboring vertices and the connecting edge weights. The goal of such an exploration is to minimize its total cost, where each edge traversal incurs a cost of the corresponding edge weight. They investigate the problem on tadpole graphs (also known as dragons, kites), which consist of a cycle with an attached path.

• The paper A tight lower bound for the capture time of the Cops and Robbers game by Sebastian Brandt, Yuval Emek, Jara Uitto, and Roger Wattenhofer proves that the upper bound O(n k+1 ) on the capture time in k-cop-win graphs is at most asymptotically tight for any constant k ≥ 2, answering in the negative an open question of Bonato and Nowakowski who asked whether this upper bound could be improved. Their result yields a surprising gap in the capture time complexities between the 1-cop and the 2-cop cases since in 1-cop-win graphs, the cop can capture the robber in O(n) time.

• Masood Masjoody and Ladislav Stacho in their paper Cops and robbers on graphs with a set of forbidden induced subgraphs characterize all sets H of graphs with bounded diameter, such that Hfree graphs are cop-bounded. This gives a characterization of cop-bounded classes of graphs defined by a finite set of connected graphs as forbidden induced subgraphs. Furthermore, they extend the characterization to the case of cop-bounded classes of graphs defined by a set H of forbidden graphs such that the components of members of H have bounded diameter.

• In the paper Simple strategies versus optimal schedules in multi-agent patrolling, the authors Akitoshi Kawamura and Makoto Soejima consider the cops and robbers game where mobile agents are assigned different maximum speeds and their goal is to patrol a fence together so as to minimize the longest time interval during which a point on the fence is left unvisited. The authors consider the case where the fence is an interval and a circle and construct schedules that are 4/3 times (for Interval Patrolling) and 21/20 times (for Circle Patrolling) as good, respectively, as the simple strategies proposed in the literature. They also propose a new variant, in which the goal is to patrol a single point under the constraint that each agent can only visit the point some predefined time after its previous visit and obtain some similar ratio bounds and NP-hardness results related to this problem.

• Sebastián González Hermosillo de la Maza, Seyyed Aliasghar Hosseini, Fiachra Knox, Bojan Mohar, and Bruce Reed in their paper Cops and robbers on oriented toroidal grids consider the straight-ahead orientations of 4-regular quadrangulations of the torus and the Klein bottle and prove that their cop number is bounded by a constant. They also show that the cop number of every k-regularly oriented toroidal grid is at most 13.

We are hopeful that the results selected for presentation in this Special Issue will contribute to further advances in the field of Graph Searching.

Guest Editors