Protection of a network by complete secure domination

Abstract

A complete secure dominating set of a graph $G$ is a dominating set $D \subseteq V(G)$ with the property that for each $v \in D$, there exists $F=\lbrace v_{j} \vert v_{j} \in N(v) \cap (V(G)-D)\rbrace$, such that for each $v_{j} \in F$, $( D-\lbrace v \rbrace) \cup \lbrace v_{j} \rbrace$ is a dominating set. The minimum cardinality of any complete secure dominating set is called the complete secure domination number of $G$ and is denoted by $\gamma_{csd}(G)$. In this paper, the bounds for complete secure domination number for some standard graphs like grid graphs and stacked prism graphs in terms of number of vertices of $G$ are found and also the bounds for the complete secure domination number of a tree are obtained in terms of different parameters of $G$.

Received: 16 May 2021 | Accepted: 19 August 2021