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20 pages, 390 KB

Open AccessArticle

Injective Hulls of Infinite Totally Split-Decomposable Metric Spaces

by Maël Pavón

Axioms 2025, 14(8), 606; https://doi.org/10.3390/axioms14080606 - 4 Aug 2025

Viewed by 1273

We extend the theory of splits in finite metric spaces to infinite ones. Within this more general framework, we investigate the class of spaces having metrics that are integer-valued and totally split-decomposable, as well as the polyhedral complex structure of their injective hulls. [...] Read more.

We extend the theory of splits in finite metric spaces to infinite ones. Within this more general framework, we investigate the class of spaces having metrics that are integer-valued and totally split-decomposable, as well as the polyhedral complex structure of their injective hulls. For this class, we provide a characterization for the injective hull to be combinatorially equivalent to a CAT(0) cube complex. Intermediate results include the generalization of the decomposition theory introduced by Bandelt and Dress in 1992 as well as results on the tight span of totally split-decomposable metric spaces proved by Huber, Koolen, and Moulton in 2006. Next, using results of Lang from 2013, we obtain proper actions on CAT(0) cube complexes for finitely generated groups endowed with a totally split-decomposable word metric and for which the associated splits satisfy a simple combinatorial property. In the case of Gromov hyperbolic groups, the obtained action is both proper aand co-compact. Finally, we obtain as an application that injective hulls of odd cycles are cell complexes isomorphic to CAT(0) cube complexes. Full article

(This article belongs to the Section Geometry and Topology)

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13 pages, 436 KB

Open AccessArticle

Graph-Theoretic Characterization of Separation Conditions in Self-Affine Iterated Function Systems

by Ming-Qi Bai, Jun Luo and Yi Wu

Fractal Fract. 2025, 9(5), 307; https://doi.org/10.3390/fractalfract9050307 - 8 May 2025

Viewed by 959

Abstract

For a self-affine iterated function system (IFS)

{f_{j}}_{j = 1}^{N}

on

R^{d}

defined by

f_{j} (x) = A^{- 1} (x + d_{j})

, where A is an expansive matrix and [...] Read more.

For a self-affine iterated function system (IFS)

{f_{j}}_{j = 1}^{N}

on

R^{d}

defined by

f_{j} (x) = A^{- 1} (x + d_{j})

, where A is an expansive matrix and

d_{j} \in R^{d}

, we reveal a novel characterization of the open set and weak separation conditions through the bounded degree of the augmented tree induced by the IFS. Furthermore, the augmented tree is shown to be a Gromov hyperbolic graph, and its hyperbolic boundary is Hölder equivalent to the self-affine set generated by the IFS, establishing a canonical association between self-affine IFSs and Gromov hyperbolic graphs. Full article

(This article belongs to the Section Geometry)

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35 pages, 599 KB

Open AccessArticle

Multi Polar q-Rung Orthopair Fuzzy Graphs with Some Topological Indices

by Andleeb Kausar, Nabilah Abughazalah and Naveed Yaqoob

Symmetry 2023, 15(12), 2131; https://doi.org/10.3390/sym15122131 - 30 Nov 2023

Cited by 1 | Viewed by 2043

Abstract

The importance of symmetry in graph theory has always been significant, but in recent years, it has become much more so in a number of subfields, including but not limited to domination theory, topological indices, Gromov hyperbolic graphs, and the metric dimension of [...] Read more.

The importance of symmetry in graph theory has always been significant, but in recent years, it has become much more so in a number of subfields, including but not limited to domination theory, topological indices, Gromov hyperbolic graphs, and the metric dimension of graphs. The purpose of this monograph is to initiate the idea of a multi polar q-rung orthopair fuzzy graphs (m-

PqROPFG)

as a fusion of multi polar fuzzy graphs and q-rung orthopair fuzzy graphs. Moreover, for a vertex of multi polar q-rung orthopair fuzzy graphs, the degree and total degree of the vertex are defined. Then, some product operations, inclusive of direct, Cartesian, semi strong, strong lexicographic products, and the union of multi polar q-rung orthopair fuzzy graphs (m-

PqROPFG s)

, are obtained. Also, at first we define some degree based fuzzy topological indices of m-

PqROPFG

. Then, we compute Zareb indices of the first and second kind, Randic indices, and harmonic index of a m-

PqROPFG

. Full article

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12 pages, 307 KB

Open AccessArticle

Gromov Hyperbolicity in Directed Graphs

by Ana Portilla, José M. Rodríguez, José M. Sigarreta and Eva Tourís

Symmetry 2020, 12(1), 105; https://doi.org/10.3390/sym12010105 - 6 Jan 2020

Cited by 1 | Viewed by 4002

Abstract

In this paper, we generalize the classical definition of Gromov hyperbolicity to the context of directed graphs and we extend one of the main results of the theory: the equivalence of the Gromov hyperbolicity and the geodesic stability. This theorem has potential applications [...] Read more.

In this paper, we generalize the classical definition of Gromov hyperbolicity to the context of directed graphs and we extend one of the main results of the theory: the equivalence of the Gromov hyperbolicity and the geodesic stability. This theorem has potential applications to the development of solutions for secure data transfer on the internet. Full article

(This article belongs to the Section B: Mathematics)

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10 pages, 247 KB

Open AccessArticle

Hyperbolicity on Graph Operators

by J. A. Méndez-Bermúdez, Rosalío Reyes, José M. Rodríguez and José M. Sigarreta

Symmetry 2018, 10(9), 360; https://doi.org/10.3390/sym10090360 - 24 Aug 2018

Cited by 8 | Viewed by 3595

Abstract

A graph operator is a mapping

F : Γ \to Γ^{'}

, where

Γ

and

Γ^{'}

are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations [...] Read more.

A graph operator is a mapping

F : Γ \to Γ^{'}

, where

Γ

and

Γ^{'}

are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph,

Λ (G)

; subdivision graph,

S (G)

; total graph,

T (G)

; and the operators

R (G)

and

Q (G)

. In particular, we get relationships such as

δ (G) \leq δ (R (G)) \leq δ (G) + 1 / 2

,

δ (Λ (G)) \leq δ (Q (G)) \leq δ (Λ (G)) + 1 / 2

,

δ (S (G)) \leq 2 δ (R (G)) \leq δ (S (G)) + 1

and

δ (R (G)) - 1 / 2 \leq δ (Λ (G)) \leq 5 δ (R (G)) + 5 / 2

for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices. Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

25 pages, 739 KB

Open AccessArticle

Hyperbolicity of Direct Products of Graphs

by Walter Carballosa, Amauris De la Cruz, Alvaro Martínez-Pérez and José M. Rodríguez

Symmetry 2018, 10(7), 279; https://doi.org/10.3390/sym10070279 - 12 Jul 2018

Cited by 2 | Viewed by 4121

Abstract

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 [...] Read more.

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} \times 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). Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

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13 pages, 776 KB

Open AccessArticle

Mathematical Properties on the Hyperbolicity of Interval Graphs

by Juan C. Hernández-Gómez, Rosalío Reyes, José M. Rodríguez and José M. Sigarreta

Symmetry 2017, 9(11), 255; https://doi.org/10.3390/sym9110255 - 1 Nov 2017

Cited by 5 | Viewed by 3990

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Graph Theory)

17 pages, 562 KB

Open AccessArticle

Generalized Chordality, Vertex Separators and Hyperbolicity on Graphs

by Álvaro Martínez-Pérez

Symmetry 2017, 9(10), 199; https://doi.org/10.3390/sym9100199 - 24 Sep 2017

Cited by 7 | Viewed by 4649

Abstract

A graph is chordal if every induced cycle has exactly three edges. A vertex separator set in a graph is a set of vertices that disconnects two vertices. A graph is

δ

-hyperbolic if every geodesic triangle is

δ

-thin. In this paper, [...] Read more.

A graph is chordal if every induced cycle has exactly three edges. A vertex separator set in a graph is a set of vertices that disconnects two vertices. A graph is

δ

-hyperbolic if every geodesic triangle is

δ

-thin. In this paper, we study the relation between vertex separator sets, certain chordality properties that generalize 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. Full article

(This article belongs to the Special Issue Graph Theory)

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20 pages, 850 KB

Open AccessFeature PaperArticle

Gromov Hyperbolicity in Mycielskian Graphs

by Ana Granados, Domingo Pestana, Ana Portilla and José M. Rodríguez

Symmetry 2017, 9(8), 131; https://doi.org/10.3390/sym9080131 - 27 Jul 2017

Cited by 5 | Viewed by 5588

Abstract

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 [...] Read more.

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

δ (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 \leq δ (G^{M}) \leq 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

δ (G)

just in terms of

δ (G^{M})

is obtained. Full article

(This article belongs to the Special Issue Graph Theory)

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