On the Complexity of Orthogonal Colouring Games and the NP-Completeness of Recognising Graphs Admitting a Strictly Matched Involution

Stephan Dominique Andres 1 François Dross 2 Melissa Huggan 3 Fionn Mc Inerney 2 Richard Nowakowski 3
2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : We consider two variants of orthogonal colouring games on graphs. In these games, two players alternate colouring vertices (from a choice of m ∈ N colours) of a pair of isomorphic graphs while respecting the properness and the orthogonality of the colouring. In the normal play variant, the player who is unable to move loses. In the scoring variant, each player aims to maximize her score which is the number of coloured vertices in the copy of the graph she owns. It is proven that, given an instance with a partial colouring, both the normal play and the scoring variant of the game are PSPACE-complete. An involution σ of a graph G is strictly matched if its fixed point set induces a clique and vσ(v) is an edge for any non-fixed point v ∈ V (G). Andres, Huggan, Mc Inerney, and Nowakowski (The orthogonal colouring game. Submitted) 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.
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https://hal.inria.fr/hal-02053265
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Submitted on : Monday, March 4, 2019 - 11:30:50 AM
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Stephan Dominique Andres, François Dross, Melissa Huggan, Fionn Mc Inerney, Richard Nowakowski. On the Complexity of Orthogonal Colouring Games and the NP-Completeness of Recognising Graphs Admitting a Strictly Matched Involution. [Research Report] Inria - Sophia Antipolis. 2019. ⟨hal-02053265v2⟩

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