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 \( \gamma_{cs}(G) \).