Location-domination and matching in cubic graphs

A dominating set of a graph G is a set D of vertices of G such that every vertex outside D is adjacent to a vertex in D. A locating-dominating set of 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 twinfree 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 of order n without isolated vertices, then γL(G) ≤ n2 . We prove the conjecture for cubic graphs. We rely heavily on proof techniques from matching theory to prove our result.