a b s t r a c tA locating-dominating set of a 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) 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 γ L (G), is the minimum cardinality of a locating-dominating set in G. It is conjectured by Garijo et al. (2014) that if G is a twin-free graph of order n without isolated vertices, then γ L (G) ≤ n 2 .We prove the general bound γ L (G) ≤ 2n 3 , slightly improving over the ⌊ 2n 3 ⌋ + 1 bound of Garijo et al. We then provide constructions of graphs reaching the n 2 bound, showing that if the conjecture is true, the family of extremal graphs is a very rich one. Moreover, we characterize the trees G that are extremal for this bound. We finally prove the conjecture for split graphs and co-bipartite graphs.

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In this paper, we continue the study of power domination in graphs (see [T. W.

We solve a number of problems posed by Hedetniemi, Hedetniemi, Laskar, Markus, and Slater concerning pairs of disjoint sets in graphs which are dominating or independent and dominating.

We prove that for graphs of order n, minimum degree δ ≥ 2 and girth g ≥ 5 the domination number γ satisfies γ ≤ n. As a corollary this implies that for cubic graphs of order n and girth g ≥ 5 the domination number γ satisfies γ ≤ 44 135n which improves recent results due to Kostochka and Stodolsky (An upper bound on the domination number of n-vertex connected cubic graphs, manuscript (2005)) and Kawarabayashi, Plummer and Saito (Domination in a graph with a 2-factor, J. Graph Theory 52 (2006), 1-6) for large enough girth. Furthermore, it confirms a conjecture due to Reed about connected cubic graphs (Paths, stars and the number three, Combin. Prob. Comput. 5 (1996), 267-276) for girth at least 83.