14 research outputs found

    Cop and robber game and hyperbolicity

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    In this note, we prove that all cop-win graphs G in the game in which the robber and the cop move at different speeds s and s' with s'<s, are \delta-hyperbolic with \delta=O(s^2). We also show that the dependency between \delta and s is linear if s-s'=\Omega(s) and G obeys a slightly stronger condition. This solves an open question from the paper (J. Chalopin et al., Cop and robber games when the robber can hide and ride, SIAM J. Discr. Math. 25 (2011) 333-359). Since any \delta-hyperbolic graph is cop-win for s=2r and s'=r+2\delta for any r>0, this establishes a new - game-theoretical - characterization of Gromov hyperbolicity. We also show that for weakly modular graphs the dependency between \delta and s is linear for any s'<s. Using these results, we describe a simple constant-factor approximation of the hyperbolicity \delta of a graph on n vertices in O(n^2) time when the graph is given by its distance-matrix

    Slimness of graphs

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    Slimness of a graph measures the local deviation of its metric from a tree metric. In a graph G=(V,E)G=(V,E), a geodesic triangle (x,y,z)\bigtriangleup(x,y,z) with x,y,zVx, y, z\in V is the union P(x,y)P(x,z)P(y,z)P(x,y) \cup P(x,z) \cup P(y,z) of three shortest paths connecting these vertices. A geodesic triangle (x,y,z)\bigtriangleup(x,y,z) is called δ\delta-slim if for any vertex uVu\in V on any side P(x,y)P(x,y) the distance from uu to P(x,z)P(y,z)P(x,z) \cup P(y,z) is at most δ\delta, i.e. each path is contained in the union of the δ\delta-neighborhoods of two others. A graph GG is called δ\delta-slim, if all geodesic triangles in GG are δ\delta-slim. The smallest value δ\delta for which GG is δ\delta-slim is called the slimness of GG. In this paper, using the layering partition technique, we obtain sharp bounds on slimness of such families of graphs as (1) graphs with cluster-diameter Δ(G)\Delta(G) of a layering partition of GG, (2) graphs with tree-length λ\lambda, (3) graphs with tree-breadth ρ\rho, (4) kk-chordal graphs, AT-free graphs and HHD-free graphs. Additionally, we show that the slimness of every 4-chordal graph is at most 2 and characterize those 4-chordal graphs for which the slimness of every of its induced subgraph is at most 1

    A simple approach for lower-bounding the distortion in any Hyperbolic embedding

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    International audienceWe answer open questions of [Verbeek and Suri, SOCG'14] on the relationships between Gromov hyperbolicity and the optimal stretch of graph embeddings in Hyperbolic space. Then, based on the relationships between hyperbolicity and Cops and Robber games, we turn necessary conditions for a graph to be Cop-win into sufficient conditions for a graph to have a large hyperbolicity (and so, no low-stretch embedding in Hyperbolic space). In doing so we derive lower-bounds on the hyperbolicity in various graph classes – such as Cayley graphs, distance-regular graphs and generalized polygons, to name a few. It partly fills in a gap in the literature on Gromov hyperbolicity, for which few lower-bound techniques are known

    On the hyperbolicity of random graphs

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    Let G=(V,E)G=(V,E) be a connected graph with the usual (graph) distance metric d:V×VN{0}d:V \times V \to N \cup \{0 \}. Introduced by Gromov, GG is δ\delta-hyperbolic if for every four vertices u,v,x,yVu,v,x,y \in V, the two largest values of the three sums d(u,v)+d(x,y),d(u,x)+d(v,y),d(u,y)+d(v,x)d(u,v)+d(x,y), d(u,x)+d(v,y), d(u,y)+d(v,x) differ by at most 2δ2\delta. In this paper, we determinate the value of this hyperbolicity for most binomial random graphs.Comment: 20 page

    Mathematical Properties on the Hyperbolicity of Interval Graphs

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    Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. In particular, we are interested in interval and indifference graphs, which are important classes of intersection and Euclidean graphs, respectively. Interval graphs (with a very weak hypothesis) and indifference graphs are hyperbolic. In this paper, we give a sharp bound for their hyperbolicity constants. The main result in this paper is the study of the hyperbolicity constant of every interval graph with edges of length 1. Moreover, we obtain sharp estimates for the hyperbolicity constant of the complement of any interval graph with edges of length 1.This paper was supported in part by a grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México and by two grants from the Ministerio de Economía y Competitividad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain. We would like to thank the referees for their careful reading of the manuscript and several useful comments that have helped us to improve the presentation of the paper

    Core congestion is inherent in hyperbolic networks

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    We investigate the impact the negative curvature has on the traffic congestion in large-scale networks. We prove that every Gromov hyperbolic network GG admits a core, thus answering in the positive a conjecture by Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which is based on the experimental observation by Narayan and Saniee, Physical Review E, 84 (2011) that real-world networks with small hyperbolicity have a core congestion. Namely, we prove that for every subset XX of vertices of a δ\delta-hyperbolic graph GG there exists a vertex mm of GG such that the disk D(m,4δ)D(m,4 \delta) of radius 4δ4 \delta centered at mm intercepts at least one half of the total flow between all pairs of vertices of XX, where the flow between two vertices x,yXx,y\in X is carried by geodesic (or quasi-geodesic) (x,y)(x,y)-paths. A set SS intercepts the flow between two nodes xx and yy if SS intersect every shortest path between xx and yy. Differently from what was conjectured by Jonckheere et al., we show that mm is not (and cannot be) the center of mass of XX but is a node close to the median of XX in the so-called injective hull of XX. In case of non-uniform traffic between nodes of XX (in this case, the unit flow exists only between certain pairs of nodes of XX defined by a commodity graph RR), we prove a primal-dual result showing that for any ρ>5δ\rho>5\delta the size of a ρ\rho-multi-core (i.e., the number of disks of radius ρ\rho) intercepting all pairs of RR is upper bounded by the maximum number of pairwise (ρ3δ)(\rho-3\delta)-apart pairs of RR

    Hyperbolicity of direct products of graphs

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    It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product G(1) x G(2) is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs).This work was supported in part by four grants from Ministerio de Economía y Competititvidad (MTM2012-30719, MTM2013-46374-P, MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain
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