$k$-STRONG DEFENSIVE ALLIANCES IN GRAPHS
Print ISSN: 0972-7752 | Online ISSN: 2582-0850 | Total Downloads : 218
DOI: https://doi.org/10.56827/SEAJMMS.2023.1901.16
Author :
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)
Abstract
In a simple connected graph $G=(V,E)$, a subset $S$ of $V$ is a defensive alliance if every vertex $v\in S$ 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 $a^{k}(G)$. 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.
Keywords and Phrases
Alliances, Strong Defensive Alliances, Defensive Alliance Number.
A.M.S. subject classification
05C69, 05C70, 05C76.
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