2025/11/30 by S, Savitha K, Klavžar, Sandi, James, Tijo
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2512.00813
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. Special attention is paid to lexicographic products in which the second factor is either complete, or a path, or a cycle. For instance, in G∘ P2ℓ and in G∘ C2ℓ, Resolver always wins, while in G∘ P2ℓ+1 and in G∘ C2ℓ+1 the same conclusion holds provided G is free from false twins. On the other hand, Spoiler always wins on G∘ P5. In most of the cases, the corresponding Maker-Breaker resolving number is also determined.