Domination and location in twin-free digraphs

A dominating set \(D\) in a digraph is a set of vertices such that every vertex is either in \(D\) or has an in-neighbour in \(D\). A dominating set \(D\) of a digraph is locating-dominating if every vertex not in \(D\) has a unique set of in-neighbours within \(D\). The location-domination number \(\gamma_L(G)\) of a digraph \(G\) is the smallest size of a locating-dominating set of \(G\). We investigate upper bounds on \(\gamma_L(G)\) in terms of the order of \(G\). We characterize those digraphs with location-domination number equal to the order or the order minus one. Such digraphs always have many twins: vertices with the same (open or closed) in-neighbourhoods. Thus, we investigate the value of \(\gamma_L(G)\) in the absence of twins and give a general method for constructing small locating-dominating sets by the means of special dominating sets. In this way, we show that for every twin-free digraph \(G\) of order \(n\), \(\gamma_L(G)\leq\frac{4n}{5}\) holds, and there exist twin-free digraphs \(G\) with \(\gamma_L(G)=\frac{2(n-2)}{3}\). If moreover \(G\) is a tournament or is acyclic, the bound is improved to \(\gamma_L(G)\leq\lceil\frac{n}{2}\rceil\), which is tight in both cases.