Locating-dominating sets: From graphs to oriented graphs

Abstract

A locating-dominating set of an undirected graph is a subset of vertices S such that S is dominating and for every u,v∉S, the neighbourhood of u and v on S are distinct (i.e. N(u)∩S≠N(v)∩S). Locating-dominating sets have received a considerable attention in the last decades. 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 each w∈V∖S, N−(w)∩S≠∅ and for each pair of distinct vertices u,v∈V∖S, N−(u)∩S≠N−(v)∩S. We consider the following two parameters. Given an undirected graph G, we look for γ→LD(G) (Γ→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 γ→LD(G)≤γLD(D)≤Γ→LD(G) where γLD(D) is the minimum size of a locating-dominating set of D.For the best orientation, we prove that, for every twin-free graph G on n vertices, γ→LD(G)≤n/2 which proves a “directed version” of a widely studied conjecture on the location-domination number. As a side result we obtain a new improved upper bound for the location-domination number in undirected trees. Moreover, we give some bounds for γ→LD(G) on many graph classes and drastically improve the value n/2 for (almost) d-regular graphs by showing that γ→LD(G)∈O(log⁡d/d⋅n) using a probabilistic argument.While γ→LD(G)≤γLD(G) holds for every graph G, we give some graph classes such as outerplanar graphs for which Γ→LD(G)≥γLD(G) and some for which Γ→LD(G)≤γLD(G) such as complete graphs. We also give general bounds for Γ→LD(G) such as Γ→LD(G)≥α(G). Finally, we show that for many graph classes Γ→LD(G) is polynomial on n but we leave open the question whether there exist graphs with Γ→LD(G)∈O(log⁡n).