Powerful alliances in graphs

Abstract

a b s t r a c tFor a graph G = (V, E), a non-empty set S ⊆ V is a defensive alliance if for every vertex v in S, v has at most one more neighbor in V − S than it has in S, and S is an offensive alliance if for every v ∈ V − S that has a neighbor in S, v has more neighbors in S than in V − S. A powerful alliance is both defensive and offensive. We initiate the study of powerful alliances in graphs.

Secure sets and their expansion in cubic graphs

“…Thus V (H) satisfies condition (2). Moreover, every vertex of H also satisfies condition (2), which implies that δ(G) ≥ 2, and the necessity follows.…”

“…Moreover, if an alliance is both offensive and defensive then we say that it is a powerful alliance [2]. Furthermore, an alliance is global if it is also a dominating set.…”

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.…”

Global defensive alliances in the lexicographic product of paths and cycles

“…Since then many variatons appeared. The most extensively studied are defensive alliances [9,8,13,19,21], offensive alliances [7,12,18] and powerful or dual alliances [2,3,22]. A more generalized concept of alliance is represented by k-alliances [1,15,16,17,19], and Dourado et al presented a new definition of alliances, namely, (f, g)-alliances [6], that generalizes previous concepts.…”