On the hyperbolicity of bipartite graphs and intersection graphs

Abstract

International audienceHyperbolicity is a measure of the tree-likeness of a graph from a metric perspective. Recently , it has been used to classify complex networks depending on their underlying geometry. Motivated by a better understanding of the structure of graphs with bounded hyperbolicity, we here investigate on the hyperbolicity of bipartite graphs. More precisely, given a bipartite graph B = (V0 ∪ V1 , E) we prove it is enough to consider any one side Vi of the bipartition of B to obtain a close approximate of its hyperbolicity δ(B) — up to an additive constant 2. We obtain from this result the sharp bounds δ(G) − 1 ≤ δ(L(G)) ≤ δ(G) + 1 and δ(G) − 1 ≤ δ(K(G)) ≤ δ(G) + 1 for every graph G, with L(G) and K(G) being respectively the line graph and the clique graph of G. Finally, promising extensions of our techniques to a broader class of intersection graphs are discussed and illustrated with the case of the biclique graph BK(G), for which we prove (δ(G) − 3)/2 ≤ δ(BK(G)) ≤ (δ(G) + 3)/2

On the hyperbolicity constant of circular-arc graphs

“…In particular, for line, clique and biclique graphs, and some extensions of line graphs (incidence, total, middle and k-edge graphs). Hence, the results in [18] are different from the results in this paper. Furthermore, the bounds in [18] refer to the hyperbolicity constant with respect to the four-point definition of hyperbolicity; the inequalities for the hyperbolicity constant with respect to a definition can be translated to the hyperbolicity constant with respect to another definition, with additional multiplicative and/or additive constants (for instance, multiplying or dividing by 3 the initial upper or lower bound, respectively).…”

“…Hence, the results in [18] are different from the results in this paper. Furthermore, the bounds in [18] refer to the hyperbolicity constant with respect to the four-point definition of hyperbolicity; the inequalities for the hyperbolicity constant with respect to a definition can be translated to the hyperbolicity constant with respect to another definition, with additional multiplicative and/or additive constants (for instance, multiplying or dividing by 3 the initial upper or lower bound, respectively). But note that it is not difficult to check that circular-arc graphs are hyperbolic; for a fixed definition of hyperbolicity, the challenge is to obtain sharp bounds for the hyperbolicity constant.…”

“…In [48] it was proved the equivalence of the hyperbolicity of many negatively curved surfaces and the hyperbolicity of a graph related to it; hence, it is useful to know hyperbolicity criteria for graphs from a geometrical viewpoint. In recent years, the study of mathematical properties of Gromov hyperbolic spaces has become a topic of increasing interest in graph theory and its applications (see, e.g., [3,7,9,15,16,17,18,20,21,26,28,30,32,42,43,44,47,48,50] and the references therein).…”

Hyperbolicity on Graph Operators

“…The hyperbolicity of the line graph has been studied previously (see [21][22][23]). We have the following results.…”

Several extremal problems on graphs involving the circumference, girth, and hyperbolicity constant

“…The study of Gromov hyperbolic graphs is a subject of increasing interest in graph theory; see, e.g., [4,11,17,18,19,21,22,23,24,27,28,30,31,34,36,46,50,52] and the references therein.…”