Offensive alliances in cubic graphs

Abstract

An offensive alliance in a graph Γ = (V, E) is a set of vertices S ⊂ V where for every vertex v in its boundary it holds that the majority of vertices in v's closed neighborhood are in S. In the case of strong offensive alliance, strict majority is required. An alliance S is called global if it affects every vertex in V \S, that is, S is a dominating set of Γ. The global offensive alliance number γ o (Γ) (respectively, global strong offensive alliance number γô(Γ)) is the minimum cardinality of a global offensive (respectively, global strong offensive) alliance in Γ. If Γ has global independent offensive alliances, then the global independent offensive alliance number γ i (Γ) is the minimum cardinality among all independent global offensive alliances of Γ. In this paper we study mathematical properties of the global (strong) alliance number of cubic graphs. For instance, we show that for all connected cubic graph of order n,

Partitioning a Graph into Global Powerful k-Alliances

“…In these papers they proposed different types of alliances: namely, defensive, offensive and dual or powerful alliances. For instance, a defensive alliance [9,10,12,14,17] of a graph is a set S of vertices of with the property that every vertex in S has at most one more neighbor outside of S than it has in S. An offensive alliance [6,14,16,17,21] of a graph is a set S of vertices of with the property that every vertex in the neighborhood of S has at least one more neighbor in S than it has outside of S. A powerful alliance [2,3,7,24] is a combination of both, defensive and offensive alliances.…”

Boundary defensive k-alliances in graphs

“…They proposed alliances of different types: namely, defensive [6][7][8][9]12,16], offensive [3,9,10,13,17] and dual or powerful alliances [1,2,18]. For instance, a defensive alliance of a graph G is a set S of vertices of G with the property that every vertex in S has at most one more neighbor outside of S than it has in S. A generalization of defensive alliances was presented by Shafique and Dutton in [14,15] where they define a defensive k-alliance as a set S of vertices of G with the property that every vertex in S has at least k more neighbors in S than it has outside of S. In this paper, we study the mathematical properties of a particular case of k-alliances that we call boundary k-alliances: we define a boundary defensive k-alliance in G as a set S of vertices of G with the property that every vertex in S has exactly k more neighbors in S than it has outside of S. We obtain several bounds on the cardinality of every boundary defensive k-alliance.…”

On global offensive k-alliances in graphs

“…They proposed different types of alliances that have been extensively studied during the last four years. These types of alliances are called defensive alliances [6,9,10,14], offensive alliances [3,11,15] and dual alliances or powerful alliances [1,8]. A generalization of these alliances called k-alliances was presented by K. H. Shafique and R. D. Dutton [12,13].…”