3,605 research outputs found
On the hyperbolicity constant in graphs
AbstractIf X is a geodesic metric space and x1,x2,x3∈X, a geodesic triangle T={x1,x2,x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if, for every geodesic triangle T in X, every side of T is contained in a δ-neighborhood of the union of the other two sides. We denote by δ(X) the sharpest hyperbolicity constant of X, i.e. δ(X)≔inf{δ≥0:X is δ-hyperbolic}. In this paper, we obtain several tight bounds for the hyperbolicity constant of a graph and precise values of this constant for some important families of graphs. In particular, we investigate the relationship between the hyperbolicity constant of a graph and its number of edges, diameter and cycles. As a consequence of our results, we show that if G is any graph with m edges with lengths {lk}k=1m, then δ(G)≤∑k=1mlk/4, and δ(G)=∑k=1mlk/4 if and only if G is isomorphic to Cm. Moreover, we prove the inequality δ(G)≤12diamG for every graph, and we use this inequality in order to compute the precise value δ(G) for some common graphs
Gromov Hyperbolicity in Mycielskian Graphs
Since the characterization of Gromov hyperbolic graphs seems a too ambitious task, there are many papers studying the hyperbolicity of several classes of graphs. In this paper, it is proven that every Mycielskian graph G(M) is hyperbolic and that delta(G(M)) is comparable to diam (G(M)). Furthermore, we study the extremal problems of finding the smallest and largest hyperbolicity constants of such graphs; in fact, it is shown that 5/4 <= delta(G(M)) <= 5/2. Graphs G whose Mycielskian have hyperbolicity constant 5/4 or 5/2 are characterized. The hyperbolicity constants of the Mycielskian of path, cycle, complete and complete bipartite graphs are calculated explicitly. Finally, information on d (G) just in terms of d (GM) is obtained.We would like to thank the referees for their valuable comments, which have improved
the paper. This work was supported in part by two grants from Ministerio de EconomÃa y Competititvidad (MTM2013-46374-P and MTM2015-69323-REDT), Spain
Cop and robber game and hyperbolicity
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
Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs
In this paper, we study Gromov hyperbolicity and related parameters, that represent how close (locally) a metric space is to a tree from a metric point of view. The study of Gromov hyperbolicity for geodesic metric spaces can be reduced to the study of graph hyperbolicity. Our main contribution in this note is a new characterization of hyperbolicity for graphs (and for complete geodesic metric spaces). This characterization has algorithmic implications in the field of large-scale network analysis, which was one of our initial motivations. A sharp estimate of graph hyperbolicity is useful, {e.g.}, in embedding an undirected graph into hyperbolic space with minimum distortion [Verbeek and Suri, SoCG\u2714]. The hyperbolicity of a graph can be computed in polynomial-time, however it is unlikely that it can be done in subcubic time. This makes this parameter difficult to compute or to approximate on large graphs. Using our new characterization of graph hyperbolicity, we provide a simple factor 8 approximation algorithm for computing the hyperbolicity of an n-vertex graph G=(V,E) in optimal time O(n^2) (assuming that the input is the distance matrix of the graph). This algorithm leads to constant factor approximations of other graph-parameters related to hyperbolicity (thinness, slimness, and insize). We also present the first efficient algorithms for exact computation of these parameters. All of our algorithms can be used to approximate the hyperbolicity of a geodesic metric space
Fast approximation and exact computation of negative curvature parameters of graphs
In this paper, we study Gromov hyperbolicity and related parameters, that
represent how close (locally) a metric space is to a tree from a metric point
of view. The study of Gromov hyperbolicity for geodesic metric spaces can be
reduced to the study of graph hyperbolicity. The main contribution of this
paper is a new characterization of the hyperbolicity of graphs. This
characterization has algorithmic implications in the field of large-scale
network analysis. A sharp estimate of graph hyperbolicity is useful, e.g., in
embedding an undirected graph into hyperbolic space with minimum distortion
[Verbeek and Suri, SoCG'14]. The hyperbolicity of a graph can be computed in
polynomial-time, however it is unlikely that it can be done in subcubic time.
This makes this parameter difficult to compute or to approximate on large
graphs. Using our new characterization of graph hyperbolicity, we provide a
simple factor 8 approximation algorithm for computing the hyperbolicity of an
-vertex graph in optimal time (assuming that the input is
the distance matrix of the graph). This algorithm leads to constant factor
approximations of other graph-parameters related to hyperbolicity (thinness,
slimness, and insize). We also present the first efficient algorithms for exact
computation of these parameters. All of our algorithms can be used to
approximate the hyperbolicity of a geodesic metric space.
We also show that a similar characterization of hyperbolicity holds for all
geodesic metric spaces endowed with a geodesic spanning tree. Along the way, we
prove that any complete geodesic metric space has such a geodesic
spanning tree. We hope that this fundamental result can be useful in other
contexts
On several extremal problems in graph theory involving gromov hyperbolicity constant
Mención Internacional en el tÃtulo de doctorIn this Thesis we study the extremal problems of maximazing and minimazing the hyperbolicity
constant on several families of graphs. In order to
properly raise our research problem, we need to introduce some important definitions and
make some remarks on the graphs we study.Programa Oficial de Doctorado en IngenierÃa MatemáticaPresidente: Elena Romera Colmenarejo.- Secretario: Ana MarÃa Portilla Ferreira.- Vocal: José MarÃa Sigarreta Almir
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