A note on secure domination number in 2K2-free graphs

Abstract

A dominating set $D$ of a graph $G$ is secure if for each vertex $v\in V\left(G\right)-D,$ $D$ contains a neighbor $u$ of $v$ such that $(D-\left\{u\right\})\cup \left\{v\right\}$ is a dominating set of $G$. The minimum cardinality of a secure dominating set in $G$ is the secure domination number of $G$ and denoted by ${\gamma }_s\left(G\right).$ A graph is $2K_2$-free if it does not contain two independent edges as an induced subgraph. Let $\alpha \left(G\right)$ denote the independence number of $G.$ Several results gave the upper bound of ${\gamma }_s\left(G\right)$ by a function of $\alpha \left(G\right).$ In this note, we shows that ${\gamma }_s\left(G\right)\le \alpha \left(G\right)+1$ for every $2K_2$-free graph $G;$ moreover, we give an example to show the bound in our result is best possible.