Small values of the hyperbolicity constant in graphs

Abstract

to my familyGracias a mis padres y hermanos por todo el apoyo y confianza desde que decidí emprender esta aventura, pero sobre todo por el infinito cariño y amor. A todos mis amigos que siempre han estado presentes de una u otra manera, gracias por ese gran apoyo moral y por sus palabras deánimo. La vida me ha dado tantos amigos que no acabaría de mencionarlos. Infinitas gracias. Conclusions Contributions of this work Open problems BibliographyNote that, if we consider a graph G whose edges have length equal to one and a graph G k obtained from G stretching out their edges until length k, then δ(G k ) = kδ(G). Therefore, all the results in this work can be generalized when the edges of the graph have length equal to k.

On the hyperbolicity constant of circular-arc graphs

“…Hence, a subgraph, say G 1 , has hyperbolicity constant 3/4 and δ(G 2 ) ≤ 3/4. Thus, Theorems 2.5 and 2.8 give that G 1 has the (3/4)-intersection property and G 2 has either the 0-or the (3/4)-intersection property, and we obtain condition (3).…”

“…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).…”

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

“…The next theorem is a well-known fact (see, e.g., [40,Theorem 8] for a proof). The following result characterizes the graphs with hyperbolicity constant 1 (see [7,Theorem 3]). The following theorems appears in [ Let us consider the complete graph with c vertices K c , and n − c graphs G 1 , .…”

Mathematical Properties on the Hyperbolicity of Interval Graphs

“…Note that the characterization of δ(G) = 5/4 in Theorem 9 is much simpler than the one in [7]. Recall that to characterize the graphs with hyperbolicity 3/2 is a very difficult task, as it was shown in ( [7] Remark 4.19). …”

“…In order to characterize from a geometric viewpoint the interval graphs with hyperbolicity constant 1, we need the following result, which is a direct consequence of Theorems 2 and 4, and ( [7] Theorem 4.14). Theorem 6.…”