Liar’s domination in graphs under some operations

Abstract

A set $S\subseteq V(G)$ is a liar's dominating set ($lds$) of graph $G$ if $|N_G[v]\cap S|\geq 2$ for every $v\in V(G)$ and $|(N_G[u]\cup N_G[v])\cap S|\geq 3$ for any two distinct vertices $u,v \in V(G)$. The liar's domination number of $G$, denoted by $\gamma_{LR}(G)$, is the smallest cardinality of a liar's dominating set of $G$. In this paper we study the concept of liar's domination in the join, corona, and lexicographic product of graphs.