The Complexity of Two Colouring Games.

We consider two variants of on graphs. In these games, two players alternate colouring uncoloured vertices (from a choice of colours) of a pair of isomorphic graphs while respecting the properness and the orthogonality of the partial colourings. In the , the first player unable to move loses. In the , each player aims to maximise their , 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 of a graph is if its fixed point set induces a clique and for any non-fixed point . 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.