Maker-Breaker resolving game played on corona products of graphs

Abstract

The Maker-Breaker resolving game is a game played on a graph G by Resolver and Spoiler. The players taking turns alternately in which each player selects a not yet played vertex of G. The goal of Resolver is to select all the vertices in a resolving set of G, while that of Spoiler is to prevent this from happening. The outcome o(G) of the game played is one of R, S, and N , where o(G) = R (resp. o(G) = S), if Resolver (resp. Spoiler) has a winning strategy no matter who starts the game, and o(G) = N , if the first player has a winning strategy. In this paper, the game is investigated on corona products G H of graphs G and H. It is proved that if o(H) ∈ {N , S}, then o(G H) = S. No such result is possible under the assumption o(H) = R. It is proved that o(GPk) = S if k = 5, otherwise o(G Pk) = R, and that o(G Ck) = S if k = 3, otherwise o(G Ck) = R. Several results are also given on corona products in which the second factor is of diameter at most 2