Ismael González Yero

On the strong metric dimension of Cartesian and direct products of graphs

Abstract Let G be a connected graph. A vertex w strongly resolves a pair u , v of vertices of G if there exists some shortest u − w path containing v or some shortest v − w path containing u . A set W of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of W . The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G . It is known that the problem of computing the strong metric dimension of a graph is NP-hard. In this paper we obtain closed formulae for the strong metric dimension of several families of Cartesian products of graphs and direct products of graphs.

On the strong metric dimension of the strong products of graphs

Abstract Let G be a connected graph. A vertex w ∈ V.G/ strongly resolves two vertices u,v ∈ V.G/ if there exists some shortest u-w path containing v or some shortest v-w path containing u. A set S of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of S. The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G. It is well known that the problem of computing this invariant is NP-hard. In this paper we study the problem of finding exact values or sharp bounds for the strong metric dimension of strong product graphs and express these in terms of invariants of the factor graphs.

The partition dimension of strong product graphs and Cartesian product graphs

Abstract Let G = ( V , E ) be a connected graph. The distance between two vertices u , v ∈ V , denoted by d ( u , v ) , is the length of a shortest u , v -path in G . The distance between a vertex v ∈ V and a subset P ⊂ V is defined as min { d ( v , x ) : x ∈ P } , and it is denoted by d ( v , P ) . An ordered partition { P 1 , P 2 , … , P t } of vertices of a graph G , is a resolving partition of G , if all the distance vectors ( d ( v , P 1 ) , d ( v , P 2 ) , … , d ( v , P t ) ) are different. The partition dimension of G is the minimum number of sets in any resolving partition of G . In this article we study the partition dimension of strong product graphs and Cartesian product graphs. Specifically, we prove that the partition dimension of the strong product of graphs is bounded below by four and above by the product of the partition dimensions of the factor graphs. Also, we give the exact value of the partition dimension of strong product graphs when one factor is a complete graph and the other one is a path or a cycle. For the case of Cartesian product graphs, we show that its partition dimension is less than or equal to the sum of the partition dimensions of the factor graphs minus one. Moreover, we obtain an upper bound on the partition dimension of Cartesian product graphs, when one factor is a complete graph.

Bondage number of grid graphs

The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than the domination number of G. Here we study the bondage number of some grid-like graphs. In this sense, we obtain some bounds or exact values of the bondage number of some strong product and direct product of two paths.

On the partition dimension of trees

Given an ordered partition ? = { P 1 , P 2 , ? , P t } of the vertex set V of a connected graph G = ( V , E ) , the partition representation of a vertex v ? V with respect to the partition ? is the vector r ( v | ? ) = ( d ( v , P 1 ) , d ( v , P 2 ) , ? , d ( v , P t ) ) , where d ( v , P i ) represents the distance between the vertex v and the set P i . A partition ? of V is a resolving partition of G if different vertices of G have different partition representations, i.e., for every pair of vertices u , v ? V , r ( u | ? ) ? r ( v | ? ) . The partition dimension of G is the minimum number of sets in any resolving partition of G . In this paper we obtain several tight bounds on the partition dimension of trees.

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 } .

Strong metric dimension of rooted product graphs

Let G be a connected graph. A vertex w strongly resolves two different vertices of G if there exists a shortest path, which contains the vertex v or a shortest path, which contains the vertex u. A set W of vertices is a strong metric generator for G if every pair of different vertices of G is strongly resolved by some vertex of W. The smallest cardinality of a strong metric generator for G is called the strong metric dimension of G. It is known that the problem of computing this invariant is NP-hard. According to that fact, in this paper we study the problem of computing exact values or sharp bounds for the strong metric dimension of the rooted product of graphs and express these in terms of invariants of the factor graphs.

The k -metric dimension of the lexicographic product of graphs

Given a simple and connected graph G = ( V , E ) , and a positive integer k , a set S ? V is said to be a k -metric generator for G , if for any pair of different vertices u , v ? V , there exist at least k vertices w 1 , w 2 , ? , w k ? S such that d G ( u , w i ) ? d G ( v , w i ) , for every i ? { 1 , ? , k } , where d G ( x , y ) denotes the distance between x and y . The minimum cardinality of a k -metric generator is the k -metric dimension of G . A set S ? V is a k -adjacency generator for G if any two different vertices x , y ? V ( G ) satisfy | ( ( N G ( x ) ? N G ( y ) ) ? { x , y } ) ? S | ? k , where N G ( x ) ? N G ( y ) is the symmetric difference of the neighborhoods of x and y . The minimum cardinality of any k -adjacency generator is the k -adjacency dimension of G . In this article we obtain tight bounds and closed formulae for the k -metric dimension of the lexicographic product of graphs in terms of the k -adjacency dimension of the factor graphs.

Boundary defensive k-alliances in graphs

We define a boundary defensive k-alliance in a graph as a set S of vertices with the property that every vertex in S has exactly k more neighbors in S than it has outside of S. In this paper we study mathematical properties of boundary defensive k-alliances. In particular, we obtain several bounds on the cardinality of every boundary defensive k-alliance. Moreover, we consider the case in which the vertex set of a graph can be partitioned into boundary alliances, showing that if a d-regular graph G of order n can be partitioned into two boundary defensive k-alliances X and Y, then |X|=|Y|=n2 and the algebraic connectivity of G is equal to d-k.

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.