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.

We prove that a cubic graph with m edges has an induced matching with at least m/9 edges. Our result generalizes a result for planar graphs due to Kang, Mnich, and Müller (Induced matchings in subcubic planar graphs, SIAM J. Discrete Math. 26 (2012) 1383-1411) and solves a conjecture of Henning and Rautenbach (Induced matchings in subcubic graphs without short cycles, to appear in Discrete Math.).

Abstract:For an integer at least 3, we prove that if G is a graph containing no two vertex-disjoint circuits of length at least , then there is a set X of at most vertices that intersects all circuits of length at least . Our result improves the bound 2 + 3 due to Birmelé, Bondy, and Reed (The Erdős-Pósa property for long circuits, Combinatorica 27 (2007),

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