Locating-dominating sets in twin-free graphs

A locating-dominating set a of graph G is a dominating set D of G with the additional property that every two distinct vertices outside D have distinct neighbors in D; that is, for distinct vertices u and v outside D, N(u) ∩D 6= N(v) ∩D where N(u) denotes the open neighborhood of u. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-domination number of G, denoted γL(G), is the minimum cardinality of a locating-dominating set in G. It is conjectured [D. Garijo, A. González and A. Márquez. Resolving sets for breaking symmetries of graphs. arXiv pre-print:1401.3686, 2014] that if G is a twin-free graph G of order n without isolated vertices, then γL(G) ≤ n2 . We prove the general bound γL(G) ≤ 2n 3 , slightly improving over the b 2n 3 c+ 1 bound of Garijo et al. We then prove the conjecture for split graphs and co-bipartite graphs. We also provide constructions of graphs reaching the n2 bound, showing that if the conjecture is true, the family of extremal graphs is a very rich one. Finally, we characterize the trees G that are extremal for this bound.