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Maker-Breaker resolving game played on corona products of graphs

  • Tijo James,
  • Sandi Klavžar,
  • Dorota Kuziak,
  • Savitha K. S,
  • Ambat Vijayakumar

摘要

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 \(\mathcal {R}\) R , \(\mathcal {S}\) S , and \(\mathcal {N}\) N , where \(o(G)=\mathcal {R}\) o ( G ) = R (resp. \(o(G)=\mathcal {S}\) o ( G ) = S ), if Resolver (resp. Spoiler) has a winning strategy no matter who starts the game, and \(o(G)=\mathcal {N}\) o ( G ) = N , if the first player has a winning strategy. In this paper, the game is investigated on corona products \(G\odot H\) G H of graphs G and H. It is proved that if \(o(H)\in \{\mathcal {N}, \mathcal {S}\}\) o ( H ) { N , S } , then \(o(G\odot H) = \mathcal {S}\) o ( G H ) = S . No such result is possible under the assumption \(o(H) = \mathcal {R}\) o ( H ) = R . It is proved that \(o(G\odot P_k) = \mathcal {S}\) o ( G P k ) = S if \(k=5\) k = 5 , otherwise \(o(G\odot P_k) = \mathcal {R}\) o ( G P k ) = R , and that \(o(G\odot C_k) = \mathcal {S}\) o ( G C k ) = S if \(k=3\) k = 3 , otherwise \(o(G\odot C_k) = \mathcal {R}\) o ( G C k ) = R . Several results are also given on corona products in which the second factor is of diameter at most 2.