62 research outputs found
Generalized chordality, vertex separators and hyperbolicity on graphs
Let be a graph with the usual shortest-path metric. A graph is
-hyperbolic if for every geodesic triangle , any side of is
contained in a -neighborhood of the union of the other two sides. A
graph is chordal if every induced cycle has at most three edges. A vertex
separator set in a graph is a set of vertices that disconnects two vertices. In
this paper we study the relation between vertex separator sets, some chordality
properties which are natural generalizations of being chordal and the
hyperbolicity of the graph. We also give a characterization of being
quasi-isometric to a tree in terms of chordality and prove that this condition
also characterizes being hyperbolic, when restricted to triangles, and having
stable geodesics, when restricted to bigons.Comment: 16 pages, 3 figure
Slimness of graphs
Slimness of a graph measures the local deviation of its metric from a tree
metric. In a graph , a geodesic triangle with
is the union of three shortest
paths connecting these vertices. A geodesic triangle is
called -slim if for any vertex on any side the
distance from to is at most , i.e. each path
is contained in the union of the -neighborhoods of two others. A graph
is called -slim, if all geodesic triangles in are
-slim. The smallest value for which is -slim is
called the slimness of . 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 of a layering partition of , (2)
graphs with tree-length , (3) graphs with tree-breadth , (4)
-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 note on isoperimetric inequalities of Gromov hyperbolic manifolds and graphs
We study in this paper the relationship of isoperimetric inequality and hyperbolicity for
graphs and Riemannian manifolds. We obtain a characterization of graphs and Riemannian
manifolds (with bounded local geometry) satisfying the (Cheeger) isoperimetric inequality, in
terms of their Gromov boundary, improving similar results from a previous work. In particular,
we prove that having a pole is a necessary condition to have isoperimetric inequality and,
therefore, it can be removed as hypothesis.First author supported in part by a Grant from Ministerio de Ciencia, Innovación y Universidades
(PGC2018-098321-B-I00), Spain. Second author supported in part by two Grants from Ministerio de
Economía y Competitividad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo
Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spain. Also, the research of
the second author was supported by the Madrid Government (Comunidad de Madrid-Spain) under the
Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in
the context of the V PRICIT (Regional Programme of Research and Technological Innovation)
Mathematical Properties on the Hyperbolicity of Interval Graphs
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
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