We investigate the complexity of finding a winning strategy for the mis\`ere version of three games played on graphs : two variants of the game $\text{ NimG}$, introduced by Stockmann in 2004 and the game $\text{ Vertex Geography}$ on both directed and undirected graphs. We show that on general graphs those three games are $\text{ PSPACE}$-Hard or Complete. For one $\text{ PSPACE}$-Hard variant of $\text{ NimG}$, we find an algorithm to compute an effective winning strategy in time $\mathcal{ O}(\sqrt{ |V(G)|}.|E(G)|)$ when $G$ is a bipartite graph.
Categories: cs.DM math.CO