2020/05/27 by Cong X. Kang, Kang, Cong X., Sandi Klavžar +5 · 1 citation
Computer Science · Engineering · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2005.13242
openalex publication_date 2020/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A set of vertices W of a graph G is a resolving set if every vertex of G is uniquely determined by its vector of distances to W. In this paper, the Maker-Breaker resolving game is introduced. The game is played on a graph G by Resolver and Spoiler who alternately select a vertex of G not yet chosen. Resolver wins if at some point the vertices chosen by him form a resolving set of G, whereas Spoiler wins if the Resolver cannot form a resolving set of G. The outcome of the game is denoted by o(G) and R\rm MB(G) (resp. S\rm MB(G)) denotes the minimum number of moves of Resolver (resp. Spoiler) to win when Resolver has the first move. The corresponding invariants for the game when Spoiler has the first move are denoted by R'\rm MB(G) and S'\rm MB(G). Invariants R\rm MB(G), R'\rm MB(G), S\rm MB(G), and S'\rm MB(G) are compared among themselves and with the metric dimension \rm dim(G). A large class of graphs G is constructed for which R\rm MB(G) > \rm dim(G) holds. The effect of twin equivalence classes and pairing resolving sets on the Maker-Breaker resolving game is described. As an application o(G), as well as R\rm MB(G) and R'\rm MB(G) (or S\rm MB(G) and S'\rm MB(G)), are determined for several graph classes, including trees, complete multi-partite graphs, grid graphs, and torus grid graphs.