Co-secure domination in Mycielski graphs

Abstract

A set $S\subseteq V(G)$ of a connected graph $G$ is a co-secure dominating set, if $S$ is a dominating set and for each $u\in S$, there exists a vertex $v\in V(G)\setminus S$, such that $v\in N(u)$ and $(S\setminus \{u\})\cup \{v\}$ is a dominating set of $G$. The minimum cardinality of the co-secure dominating set in a graph $G$ is the co-secure domination number, ${\gamma }_{cs}(G)$. In this paper, we characterise the Mycielski graphs with co-secure domination 2 and 3. We also obtained a sharp upper bound for ${\gamma }_{cs}(G)$.