Towards a Trichotomy for Quantified H-Coloring

Hell and Nesetřil proved that the H-colouring problem is NP-complete if, and only if, H is bipartite. In this paper, we investigate the complexity of the quantified H-colouring problem (a restriction of the quantified constraint satisfaction problem to undirected graphs). We introduce this problem using a new two player colouring game. We prove that the quantified H-colouring problem is: 1. tractable, if H is bipartite; 2. NP-complete, if H is not bipartite and not connected; and, 3. Pspace-complete, if H is connected and has a unique cycle, which is of odd length. We conjecture that the last case extends to all non-bipartite connected graphs.

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