Bounds on Co-Independent Liar’s Domination in Graphs

Abstract

A set $S\subseteq V$ of a graph $G=\left(V,E\right)$ is called a co-independent liar’s dominating set of $G$ if (i) for all $v\in V$ , $\left|N_G\left[v\right]\cap S\right|\ge 2$ , (ii) for every pair $u,v\in V$ of distinct vertices, $\left|\left(N_G\left[u\right]\cup N_G\left[v\right]\right)\cap S\right|\ge 3$ , 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 ${\gamma }_{\text{coi}}^{LR}\left(G\right)$ . 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.