Domination Parameters in Shadow Graph and Path Connected Graph

Abstract

A locating-dominating set (LDS) $S$ of a graph $G$ is a dominating set $S$ of $G$ such that for every two vertices $u$ and $v$ in $V(G) \setminus S$, $N(u)\cap S \neq N(v)\cap S$. The locating-domination number $\gamma^{L}(G)$ is the minimum cardinality of a LDS of $G$. Further, if $S$ is a total dominating set then $S$ is called a locating-total dominating set (LTDS). In this paper, we determine the locating-domination and locating-total domination numbers of the shadow graph and for a special class of path connected graph.