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 Economía y Competititvidad (MTM2013-46374-P and MTM2015-69323-REDT), Spain

Graph Theory

This book contains the successful invited submissions [1–10] to a special issue of Symmetry on the subject area of ‘graph theory’ [...

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view

Hyperbolicity of direct products of graphs

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) x 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).This work was supported in part by four grants from Ministerio de Economía y Competititvidad (MTM2012-30719, MTM2013-46374-P, MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain

Gromov hyperbolicity in directed graphs

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 internetSupported in part by two grants from Ministerio de Economía y Competititvidad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spai

Gromov hiperbolicity in graphs

If 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 any side of T is contained in the δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) := inf{δ ≥ 0 : X is δ-hyperbolic } . The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. One of the main aims of this PhD Thesis is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph G \ e obtained from the graph G by deleting an arbitrary edge e from it. These inequalities allow to obtain other main result, which characterizes in a quantitative way the hyperbolicity of any graph in terms of local hyperbolicity. In this work we also obtain information about the hyperbolicity constant of the line graph L(G) in terms of properties of the graph G. In particular, we prove qualitative results as the following: a graph G is hyperbolic if and only if L(G) is hyperbolic; if {Gn} is a T-decomposition of G ({Gn} are simple subgraphs of G), the line graph L(G) is hyperbolic if and only if supn δ(L(Gn)) is finite. Besides, we obtain quantitative results when k is the length of the edges of G and L(G). Two of them are quantitative versions of our qualitative results. We also prove that g(G)/4 ≤ δ(L(G)) ≤ c(G)/4 + 2k, where g(G) is the girth of G and c(G) is its circumference. We show that δ(L(G)) ≥ sup{L(g) : g is an isometric cycle in G}/4. Besides, we obtain bounds for δ(G) + δ(L(G)). Also, we characterize the graphs G with δ(L(G)) < k. Furthermore, we consider G with edges of arbitrary lengths, and L(G) with edges of non-constant lengths. In particular, we prove that a cycle of the graph G is transformed isometrically into a cycle of the graph L(G) with the same length. We also prove that δ(G) ≤ δ(L(G)) ≤ 5δ(G) + 3lmax, where lmax := supe∈E(G) L(e). This result implies the monotony of the hyperbolicity constant under a non-trivial transformation (the line graph of a graph). Also, we obtain criteria which allow us to decide, for a large class of graphs, whether they are hyperbolic or not. We are especially interested in the planar graphs which are the “boundary” (the 1-skeleton) of a tessellation of the Euclidean plane. Furthermore, we prove that a graph obtained as the 1-skeleton of a general CW 2-complex is hyperbolic if and only if its dual graph is hyperbolic. One of the main problems on this subject is to relate the hyperbolicity with other properties on graph theory. We extend in two ways (edge-chordality and path-chordality) the classical definition of chordal graphs in order to relate this property with Gromov hyperbolicity. In fact, we prove that every edge-chordal graph is hyperbolic and that every hyperbolic graph is path-chordal. Furthermore, we prove that every path-chordal cubic graph (with small path-chordality constant) is hyperbolic. Some previous works characterize the hyperbolic product graphs (for the Cartesian product, strong product and lexicographic product) in terms of properties of the factor graphs. Finally, we characterize the hyperbolic product graphs for two important kinds of products: the graph join G1 ⊎G2 and the corona G1 ⋄G2. The graph join G1 ⊎G2 is always hyperbolic, and G1 ⋄ G2 is hyperbolic if and only if G1 is hyperbolic. Furthermore, we obtain simple formulae for the hyperbolicity constant of the graph join G1 ⊎ G2 and the corona G1 ⋄ G2. ------------------------------------------------------------------------------Sea X un espacio métrico geodésico y x1, x2, x3 ∈ X. Un triángulo geodésico T = {x1, x2, x3} es la unión de tres geodésicas [x1x2], [x2x3] y [x3x1] de X. El espacio X es δ- hiperbólico (en el sentido de Gromov) si todo lado de T está contenido en la δ-vecindad de la unión de los otros dos lados, para todo triángulo geodésico T de X. Se denota por δ(X) a la constante de hiperbolicidad óptima de X, es decir, δ(X) := inf{δ ≥ 0 : X es δ-hiperbólico }. El estudio de los grafos hiperbólicos es un tema interesante dado que la hiperbolicidad de un espacio métrico geodésico es equivalente a la hiperbolicidad de un grafo más sencillo asociado al espacio. Uno de los principales objetivos de esta Tesis Doctoral es obtener información cuantitativa sobre la variación de la constante de hiperbolicidad del grafo G \ e que se obtiene del grafo G mediante la eliminación de una arista arbitraria e de él. Estas desigualdades permiten caracterizar, de forma cuantitativa, la hiperbolicidad de cualquier grafo en términos de su hiperbolicidad local. En esta memoria se obtene información acerca de la constante de hiperbolicidad del grafo línea L(G) en términos de propiedades del grafo G. En particular, se obtienen los siguientes resultados cualitativos: un grafo G es hiperbólico si y sólo si L(G) es hiperbólico; si {Gn} es una T-descomposición de G, el grafo línea L(G) es hiperbólico si y sólo si supn δ(L(Gn)) es finito. Además, se obtienen resultados cuantitativos cuando las aristas de G y L(G) tienen longitud k. Se demuestra que g(G)/4 ≤ δ(L(G)) ≤ c(G)/4 + 2k, donde g(G) es el cuello de G y c(G) es su circunferencia. También se prueba que δ(L(G)) ≥ sup{L(g) : g es un ciclo isométrico de G}/4. Igualmente, se obtienen cotas para δ(G) + δ(L(G)) y se caracterizan los grafos G tales que δ(L(G)) < k. Por otra parte, se consideran grafos G con aristas de longitudes arbitrarias, y L(G) con aristas de longitudes no constante. En particular, se demuestra que los ciclos de G se transforman isométricamente en ciclos de L(G) con la misma longitud. También se obtiene la relación δ(G) ≤ δ(L(G)) ≤ 5δ(G) + 3lmax, donde lmax := supe∈E(G) L(e). Este resultado prueba la monotonía de la constante de hiperbolicidad bajo una transformación no trivial (el grafo línea de un grafo). También, se obtienen criterios que permiten decidir, para una clase amplia de grafos, cuando son estos hiperbólicos o no. Se presta especial atención en los grafos planares que son la “frontera” (el 1-esqueleto) de una teselación del plano euclídeo. Además, se prueba que los grafos que se obtienen como 1-esqueleto de un CW 2-complejo general son hiperbólicos si y sólo si su grafo dual es hiperbólico. Uno de los principales problemas en este área es relacionar la hiperbolicidad de un grafo con otras propiedades de la teoría de grafos. En este trabajo se extiende de dos maneras (arista-cordalidad y camino-cordalidad) la definición clásica de cordalidad con el fin de relacionar esta propiedad con la hiperbolicidad. De hecho, se demuestra que todo grafo arista-cordal es hiperbólico y que todo grafo hiperbólico es camino-cordal. También, se demuestra que todo grafo cúbico camino-cordal (con constante pequeña de camino-cordalidad) es hiperbólico. Algunos trabajos previos caracterizan la hiperbolicidad de productos de grafos (para el producto cartesiano, el producto fuerte y el producto lexicográfico) en términos de propiedades de los grafos factores. En el último capítulo, se caracteriza la hiperbolicidad de dos productos de grafos: el grafo join G1 ⊎ G2 y el corona G1 ⋄ G2. El grafo join G1 ⊎ G2 siempre es hiperbólico, y el corona G1 ⋄ G2 es hiperbólico si y sólo si G1 es hiperbólico. Además, obtenemos fórmulas sencillas para la constante de hiperbolicidad del grafo join G1 ⊎ G2 y del corona G1 ⋄ G2