A set P of vertices of a graph G is convex if it contains all vertices along shortest paths between vertices in P. The convex hull of P is the smallest convex set containing P. We say that a subset of vertices P generates the graph G if the convex hull of P is the entire vertex set. We study two impartial games Generate and Do Not Generate in which two players alternately take turns selecting previously-unselected vertices of a finite graph G. The first player who builds a generating set for the graph from the jointly-selected elements wins the achievement game GEN(G). The first player who cannot select a vertex without building a generating set loses the avoidance game DNG(G). Similar games have been considered by several authors, including Harary et al. In this talk, we determine the nim-number for several graph families, including trees, cycle graphs, complete graphs, complete bipartite graphs, and hypercube graphs.

Joint work with Bret Benesh, Marie Meyer, Sarah Salmon, and Nandor Sieben.

This talk was given on January 25, 2023 during the Combinatorial Game Theory Colloquia in S. Miguel, Azores.

April 27, 2023