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arXiv · 1408.5412

On the Complexity of Role Colouring Planar Graphs, Trees and Cographs

Christopher Purcell·M. Puck Rombach
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Abstract

We prove several results about the complexity of the role colouring problem. A role colouring of a graph $G$ is an assignment of colours to the vertices of $G$ such that two vertices of the same colour have identical sets of colours in their neighbourhoods. We show that the problem of finding a role colouring with $1< k <n$ colours is NP-hard for planar graphs. We show that restricting the problem to trees yields a polynomially solvable case, as long as $k$ is either constant or has a constant difference with $n$, the number of vertices in the tree. Finally, we prove that cographs are always $k$-role-colourable for $1<k\leq n$ and construct such a colouring in polynomial time.

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Christopher Purcell, M. Puck Rombach. 2014-08-14. On the Complexity of Role Colouring Planar Graphs, Trees and Cographs. https://arxiv.org/abs/1408.5412

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