The Complexity of Two Colouring Games

We consider two variants of orthogonal colouring games on graphs. In these games, two players alternate colouring uncoloured vertices (from a choice of [Formula: see text] colours) of a pair of isomorphic graphs while respecting the properness and the orthogonality of the partial colourings. In the normal play variant, the first player unable to move loses. In the scoring variant, each player aims to maximise their score, which is the number of coloured vertices in their copy of the graph. We prove that, given an instance with partial colourings, both the normal play and the scoring variant of the game are PSPACE-complete. An involution [Formula: see text] of a graph G is strictly matched if its fixed point set induces a clique and [Formula: see text] for any non-fixed point [Formula: see text] . Andres et al. (Theor Comput Sci 795:312–325, 2019) gave a solution of the normal play variant played on graphs that admit a strictly matched involution. We prove that recognising graphs that admit a strictly matched involution is NP-complete.