The Closed Geodetic Game: algorithms and strategies

The geodetic closure of a set S of vertices of a graph is the set of all vertices in shortest paths between pairs of vertices of S. A set S of vertices in a graph is geodetic if its geodetic closure contains all the vertices of the graph. Buckley introduced in 1984 the idea of a game where two players construct together a geodetic set by alternately selecting vertices, the game ending when all vertices are in the geodetic closure. The Geodetic Game was then studied in 1985 by Buckley and Harary, and allowed players to select vertices already in the geodetic closure of the current set. We study the more natural variant, also introduced in 1985 by Buckley and Harary and called the Closed Geodetic Game, where the players alternate adding to a set S vertices that are not in the geodetic closure of S, until no move is available. This variant was only studied ever since for trees by Araujo et al. in 2024. We provide a full characterization of the Sprague-Grundy values of graph classes such as paths and cycles, of the outcomes of several products of graphs in function of the outcomes of the two graphs, and give polynomial-time algorithms to determine the Sprague-Grundy values of cactus and block graphs.