Location-domination and matching in cubic graphs

Abstract

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≠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. Garijo et al. (2014) posed the conjecture that for n sufficiently large, the maximum value of the location-domination number of a twin-free, connected graph on n vertices is equal to ⌊n2⌋. We propose the related (stronger) conjecture 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.