Bounds on co-independent liar's domination in graphs (Q2034485)

Summary: A set \(S\subseteq V\) of a graph \(G=(V,E)\) is called a co-independent liar's dominating set of \(G\) if (i) for all \(v\in V\), \(|N_G [v] \cap S|\geq 2\), (ii) for every pair \(u\), \(v\in V\) of distinct vertices, \(|(N_G [u] \cup N_G [v]) \cap S|\geq 3\), and (iii) the induced subgraph of \(G\) on \(V-S\) has no edge. The minimum cardinality of vertices in such a set is called the co-independent liar's domination number of \(G\), and it is denoted by \(\gamma_{\text{coi}}^{LR} (G)\). In this paper, we introduce the concept of co-independent liar's domination number of the middle graph of some standard graphs such as path and cycle graphs, and we propose some bounds on this new parameter.