Locating-dominating sets: from graphs to oriented graphs

TL;DR

This paper extends the concept of locating-dominating sets from undirected to oriented graphs, providing bounds, probabilistic results, and open questions on the parameters' behavior across various graph classes.

Contribution

It introduces and analyzes the oriented version of locating-dominating sets, establishing bounds and probabilistic results, and explores their relationships with undirected graph parameters.

Findings

For twin-free graphs, the directed locating-dominating set size is at most half the vertices.

Bounds are provided for various graph classes, especially for regular graphs.

Open questions remain on the existence of graphs with logarithmic oriented locating-dominating set size.

Abstract

A locating-dominating set in an undirected graph is a subset of vertices $S$ such that $S$ is dominating and for every $u,v \notin S$, we have $N(u)\cap S\ne N(v)\cap S$. In this paper, we consider the oriented version of the problem. A locating-dominating set in an oriented graph is a set $S$ such that for every $w\in V$, $N[w]^-\cap S=\emptyset$ and for each pair of vertices $u,v\in V\setminus S$, $N^-(u)\cap S\ne N^-(v)\cap S$. We consider the following two parameters. Given an undirected graph $G$, we look for $\overset{\rightarrow}{\gamma}_{LD}(G)$ ($\overset{\rightarrow}{\Gamma}_{LD}(G))$ which is the size of the smallest (largest) optimal locating-dominating set over all orientations of $G$. In particular, if $D$ is an orientation of $G$, then $\overset{\rightarrow}{\gamma}_{LD}(G)\leq{\gamma}_{LD}(D)\leq\overset{\rightarrow}{\Gamma}_{LD}(G)$. For the best orientation, we prove that, for every twin-free graph $G$ on $n$ vertices, $\overset{\rightarrow}{\gamma}_{LD}(G)\le n/2$ proving a ``directed version'' of a conjecture on $\gamma_{LD}(G)$. Moreover, we give some bounds for $\overset{\rightarrow}{\gamma}_{LD}(G)$ on many graph classes and drastically improve the value $n/2$ for (almost) $d$-regular graphs by showing that $\overset{\rightarrow}{\gamma}_{LD}(G)\in O(\log d/d\cdot n)$ using a probabilistic argument. While $\overset{\rightarrow}{\gamma}_{LD}(G)\leq\gamma_{LD}(G)$ holds for every graph $G$, we give some graph classes graphs for which $\overset{\rightarrow}{\Gamma}_{LD}(G)\geq{\gamma}_{LD}(G)$ and some for which $\overset{\rightarrow}{\Gamma}_{LD}(G)\leq {\gamma}_{LD}(G)$. We also give general bounds for $\overset{\rightarrow}{\Gamma}_{LD}(G)$. Finally, we show that for many graph classes $\overset{\rightarrow}{\Gamma}_{LD}(G)$ is polynomial on $n$ but we leave open the question whether there exist graphs with $\overset{\rightarrow}{\Gamma}_{LD}(G)\in O(\log n)$.