k-STRONG DEFENSIVE ALLIANCES IN GRAPHS

Karthik K; Chandru Hegde; B. Sooryanarayana

doi:10.56827/SEAJMMS.2023.1901.16

Karthik K Department of Mathematics, Mangalore University, Mangalagangothri - 574199, Karnataka, INDIA
Chandru Hegde Department of Mathematics, Mangalore University, Mangalagangothri - 574199, Karnataka, INDIA
B. Sooryanarayana Department of Mathematics, Dr. Ambedkar Institute of Technology, Bengaluru - 560056, Karnataka, INDIA

DOI: https://doi.org/10.56827/SEAJMMS.2023.1901.16

Keywords: Alliances, Strong Defensive Alliances, Defensive Alliance Number.

Abstract

In a simple connected graph G = (V,E), a subset S of V is a defensive alliance if every vertex v ∈ S has at most one more neighbour in V − S than it has in S. The minimum cardinality of a defensive alliance in G is called the defensive alliance number of G, denoted by a(G). A k-strong defensive alliance S is a defensive alliance in G, in which removal of any set of at most k vertices does not affect its defensive property. The k-strong defensive alliance number of G is the minimum cardinality of a k-strong defensive alliance in G, denoted by ak(G). In this paper, some properties of k-strong defensive alliances are discussed and the k-strong defensive alliance numbers of some classes of graphs are obtained.