An extension of locating-total domination problem and its complexity

Abstract

An $r$-dominating set ($r$-total dominating set) of $G$ is a subset $S$ of $V(G)$ for which $N_{r}^{}(u)\cap S$ is non-empty for all $u$ not in $S$ (for all $u$ in $V(G)$).   %     An $r$-locating-dominating set ($r$-locating-total dominating set) of $G$ is an $r$-dominating set ($r$-total dominating set) $S$ of $G$ for which $N_{r}^{}(u)\cap S$ is different from $N_{r}^{}(v) \cap S$ for all $u$ and $v$ not in $S$.     %An $r$-dominating set ($r$-total dominating set) $S$ in a graph $G$ is called an $r$-locating-dominating set ($r$-locating-total dominating set) if for all $u$ and $v$ in $V(G) \setminus S$, $N_{r}^{}(u)\cap S$ is different from $N_{r}^{}(v) \cap S$.     %     This paper presents an extension of the locating-total dominating set of $G$.     %In this paper, we present an extension of locating-total dominating set of $G$ that we refer to as the $r$-locating-total dominating set of $G$.     Further, we establish a lower bound on $r$-locating-dominating set and $r$-locating-total dominating set for $k$-regular graphs, as well as demonstrate that $r$-locating-total dominating set is an NP-complete problem.     Furthermore, the $r$-locating-dominating set and $r$-locating-total dominating set problems are discussed for some standard graphs.