Sergio Bermudo

Gromov hyperbolic graphs

Abstract In this paper we prove that the study of the hyperbolicity on graphs can be reduced to the study of the hyperbolicity on simpler graphs. In particular, we prove that the study of the hyperbolicity on a graph with loops and multiple edges can be reduced to the study of the hyperbolicity in the same graph without its loops and multiple edges; we also prove that the study of the hyperbolicity on an arbitrary graph is equivalent to the study of the hyperbolicity on a 3-regular graph obtained from it by adding some edges and vertices. Moreover, we study how the hyperbolicity constant of a graph changes upon adding or deleting finitely or infinitely many edges.

Lower bounds on the differential of a graph

Abstract Let G = ( V , E ) be a graph of order n and let B ( D ) be the set of vertices in V ∖ D that have a neighbor in a set D . The differential of a set D is defined as ∂ ( D ) = | B ( D ) | − | D | and the differential of a graph to equal the maximum value of ∂ ( D ) for any subset D of V . In this paper, we obtain several tight lower bounds for the differential of a graph.

Small values of the hyperbolicity constant in graphs

If X is a geodesic metric space and x 1 , x 2 , x 3 ź X , a geodesic triangle T = { x 1 , x 2 , x 3 } is the union of the three geodesics x 1 x 2 , x 2 x 3 and x 3 x 1 in X . The space X is ź -hyperbolic (in the Gromov sense) if any side of T is contained in a ź -neighborhood of the union of the two other sides, for every geodesic triangle T in X . We denote by ź ( X ) the sharpest hyperbolicity constant of X , i.e., ź ( X ) : = inf { ź ź 0 : X źisź ź -hyperbolic } . In the study of any parameter on graphs it is natural to study the graphs for which this parameter has small values. In this paper we study the graphs with small hyperbolicity constant, i.e., the graphs which are like trees (in the Gromov sense). We obtain simple characterizations of the graphs G with ź ( G ) = 1 and ź ( G ) = 5 4 (the case ź ( G ) < 1 is known). Also, we give a necessary condition in order to have ź ( G ) = 3 2 (we know that ź ( G ) is a multiple of 1 4 ). Although it is not possible to obtain bounds for the diameter of graphs with small hyperbolicity constant, we obtain such bounds for the effective diameter if ź ( G ) < 3 2 . This is the best possible result, since we prove that it is not possible to obtain similar bounds if ź ( G ) ź 3 2 .

Combinatorics for smaller kernels: The differential of a graph

Let G=(V,E) be a graph of order n and let B(D) be the set of vertices in V∖D that have a neighbor in the vertex set D. The differential of D is defined as ∂(D)=|B(D)|−|D| and the differential of a graph G, written ∂(G), is equal to max⁡{∂(D):D⊆V}. If G is connected and n≥3, ∂(G)≥n/5 is known. This immediately leads to a linear vertex kernel result (in the terminology of parameterized complexity) for the problem of deciding whether ∂(G)≥k, taking k as the parameter. We then establish a new combinatorial result which establishes that ∂(G)≥n/4 if G is a connected graph of order n≥6 and if G contains no induced path of five vertices whose midpoint is a cut vertex and whose endpoints have degree one. This technical combinatorial theorem can be used to derive an even smaller linear vertex kernel for general graphs. Also, we show that the related maximization problem allows for a polynomial-time factor-14 approximation algorithm.

Computing the differential of a graph: Hardness, approximability and exact algorithms

We are studying computational complexity aspects of the differential of a graph, a graph parameter previously introduced to model ways of influencing a network. We obtain NP hardness results also for very special graph classes, such as split graphs and cubic graphs. This motivates to further classify this problem in terms of approximability. Here, one of our results shows MAXSNP completeness for the corresponding maximization problem on subcubic graphs. Moreover, we also provide a Measure & Conquer analysis for an exact moderately exponential-time algorithm that computes that graph parameter in time O(1.755^n) on a graph of order n.

On the complement graph and defensive k-alliances

In this paper, we obtain several tight bounds of the defensive k-alliance number in the complement graph from other parameters of the graph. In particular, we investigate the relationship between the alliance numbers of the complement graph and the minimum and maximum degree, the domination number and the isoperimetric number of the graph. Moreover, we prove the NP-completeness of the decision problem underlying the defensive k-alliance number.

Mathematical Properties of Gromov Hyperbolic Graphs

In this paper we deal with Gromov hyperbolic graphs. We obtain several tight bounds for the hyperbolicity constant of a graph. In particular, we investigate the relationship between the hyperbolicity constant of a graph and its edge number. We prove that the study of the hyperbolicity on graphs can be reduced to the study of the hyperbolicity on simpler graphs. In particular, we show that the study of the hyperbolicity on a graph with loops and multiple edges can be reduced to the study of the hyperbolicity in its subjacent simple graph; we also prove that the study of the hyperbolicity of an arbitrary graph is equivalent to the study of the hyperbolicity of a 3‐regular graph obtained from it by adding some edges and vertices.

Partitioning a graph into offensive k -alliances

An offensive k -alliance in a graph is a set S of vertices with the property that every vertex in the boundary of S has at least k more neighbors in S than it has outside of S . An offensive k -alliance S is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) offensive k -alliances. The (global) offensive k -alliance partition number of a graph ? = ( V , E ) , denoted by ( ? k g o ( ? ) ) ? k o ( ? ) , is defined to be the maximum number of sets in a partition of V such that each set is an offensive (a global offensive) k -alliance. We show that 2 ? ? k g o ( ? ) ? 3 - k if ? is a graph without isolated vertices and k ? { 2 - Δ , . . . , 0 } . Moreover, we show that if ? is partitionable into global offensive k -alliances for k ? 1 , then ? k g o ( ? ) = 2 . As a consequence of the study we show that the chromatic number of ? is at most 3 if ? 0 g o ( ? ) = 3 . In addition, for k ? 0 , we obtain tight bounds on ? k o ( ? ) and ? k g o ( ? ) in terms of several parameters of the graph including the order, size, minimum and maximum degree. Finally, we study the particular case of the cartesian product of graphs, showing that ? k o ( ? 1 i? ? 2 ) ? ? k 1 o ( ? 1 ) ? k 2 o ( ? 2 ) , for k ? min { k 1 - Δ 2 , k 2 - Δ 1 } , where Δ i denotes the maximum degree of ? i , and ? k g o ( ? 1 i? ? 2 ) ? max { ? k 1 g o ( ? 1 ) , ? k 2 g o ( ? 2 ) } , for k ? min { k 1 , k 2 } .

On global offensive k-alliances in graphs

Abstract We investigate the relationship between global offensive k -alliances and some characteristic sets of a graph including r -dependent sets, τ -dominating sets and standard dominating sets. In addition, we discuss the close relationships that exist among the (global) offensive k i -alliance number of Γ i , i ∈ { 1 , 2 } , and the (global) offensive k -alliance number of Γ 1 × Γ 2 , for some specific values of k . As a consequence of the study, we obtain bounds on the global offensive k -alliance number in terms of several parameters of the graph.

On the total k-domination in graphs

Abstract Let G = (V, E) be a graph; a set S ⊆ V is a total k-dominating set if every vertex v ∈ V has at least k neighbors in S. The total k-domination number γkt(G) is the minimum cardinality among all total k-dominating sets. In this paper we obtain several tight bounds for the total k-domination number of a graph. In particular, we investigate the relationship between the total k-domination number of a graph and the order, the size, the girth, the minimum and maximum degree, the diameter, and other domination parameters of the graph.