PRO-SECURE DOMINATION IN CARTESIAN AND STRONG PRODUCT OF TWO GRAPHS

Abstract

Let $G=(V(G), E(G))$ be a connected, finite and undirected graph. A set $D \subseteq V(G)$ is a pro-secure dominating set (PSDS) of a graph $G$, if $D$ is a dominating set of $G$ and for each $u \in V(G) \backslash D \quad \exists$ a vertex $v \in D$ such that $u v \in E(G)$ and $(D \backslash\{v, w\}) \cup\{u\}$ or $(D \backslash\{v, w\}) \cup \{u, z\}$ (where $w \in D$ and $z \in V(G) \backslash D$ ) is a dominating set of graph $G$. The minimum cardinality of pro-secure dominating set of graph $G$ is called the pro-secure domination number and it is denoted by $\gamma_{p r s}(G)$. In this paper, we investigate exact values of pro-secure domination number in Cartesian product as well as strong product of two graphs.

Received: June 30, 2025
Accepted: August 28, 2025