Endgame problems of Sim-like graph Ramsey avoidance games are PSPACE-complete

Abstract

In Sim, two players compete on a complete graph of six vertices (K 6) . The players alternate in coloring one as yet uncolored edge using their color. The player who first completes a monochromatic triangle (K 3) loses. Replacing K 6 and K 3 by arbitrary graphs generalizes Sim to graph Ramsey avoidance games. Given an endgame position in these games, the problem of deciding whether the player who moves next has a winning strategy is shown to be PSPACE-complete. It can be reduced from the problem of whether the first player has a winning strategy in the game G pos ( POS CNF ) (Schaefer, J. Comput. System Sci. 16 (2) (1978) 185–225). The following game variants are also shown to have PSPACE-complete endgame problems: (1) completing a monochromatic subgraph isomorphic to A is forbidden and the player who is first unable to move loses, (2) both players are allowed to color one or more edges in each move, (3) more than two players take part in the game, and (4) each player has to avoid a separate graph. In all results, the graphs to be avoided can be restricted to the bowtie graph ( ⋈, i.e., two triangles with one common vertex).