Maker–Breaker Resolving Game Played on Lexicographic Products of Graphs

In the Maker–Breaker resolving game, two players named Resolver and Spoiler alternately select unplayed vertices of a given graph G . The aim of Resolver is to select all the vertices of some resolving set of G , while Spoiler aims to select at least one vertex from every resolving set of G . In this paper, this game is investigated on the lexicographic product of graphs. It is proved that if Spoiler has a winning strategy on a graph H no matter who starts the game, or if the first player has a winning strategy on H , then Spoiler always has a winning strategy on G∘ H G ∘ H . Special attention is paid to lexicographic products in which the second factor is a complete graph, a path, or a cycle. For instance, in G∘ P₂ G ∘ P 2 ℓ and in G∘ C₂ G ∘ C 2 ℓ , Resolver always wins, while in G∘ P2 +1 G ∘ P 2 ℓ + 1 and in G∘ C2 +1 G ∘ C 2 ℓ + 1 the same conclusion holds provided G is free from false twins. On the other hand, Spoiler always wins on G∘ P₅ G ∘ P 5 . In most of the cases, the corresponding Maker-Breaker resolving number is also determined.