Locating-dominating sets: from graphs to oriented graphs
Résumé
A locating-dominating set of an undirected graph is a subset of vertices $S$ such that $S$ is dominating and for every $u,v \notin S$, the neighbourhood of $u$ and $v$ on $S$ are distinct (i.e. $N(u) \cap S \ne N(v) \cap S$). In this paper, we consider the oriented version of the problem. A locating-dominating set of an oriented graph is a set $S$ such that for every $u,v \in V \setminus S$, $N^-(u) \cap S \ne N^-(v) \cap S$. 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 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 $\overset{\rightarrow}{\Gamma}_{LD}(G)\in O(\log n)$.
Domaines
Combinatoire [math.CO]
Origine : Fichiers produits par l'(les) auteur(s)