Maker-Breaker metric resolving games on graphs

Abstract

Let [Formula: see text] denote the length of a shortest path between vertices [Formula: see text] and [Formula: see text] in a graph [Formula: see text] with vertex set [Formula: see text]. For a positive integer [Formula: see text], let [Formula: see text] and [Formula: see text]. A set [Formula: see text] is a distance-k resolving set of [Formula: see text] if [Formula: see text] for distinct [Formula: see text]. In this paper, we study the maker-breaker distance-[Formula: see text] resolving game (MB[Formula: see text]RG) played on a graph [Formula: see text] by two players, Maker and Breaker, who alternately select a vertex of [Formula: see text] not yet chosen. Maker wins by selecting vertices which form a distance-[Formula: see text] resolving set of [Formula: see text], whereas Breaker wins by preventing Maker from winning. We denote by [Formula: see text] the outcome of MB[Formula: see text]RG. Let [Formula: see text], [Formula: see text] and [Formula: see text], respectively, denote the outcome for which Maker, Breaker, and the first player has a winning strategy in MB[Formula: see text]RG. Given a graph [Formula: see text], the parameter [Formula: see text] is a non-decreasing function of [Formula: see text] with codomain [Formula: see text]. We exhibit pairs [Formula: see text] and [Formula: see text] such that the ordered pair [Formula: see text] realizes each member of the set [Formula: see text]; we provide graphs [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text]. Moreover, we obtain some general results on MB[Formula: see text]RG and study the MB[Formula: see text]RG played on some graph classes.