Efficient Algorithms for Clique-Colouring and Biclique-Colouring Unichord-Free Graphs

Abstract

The class of unichord-free graphs was recently investigated in the context of vertex-colouring (Trotignon and Vuskovic in J Graph Theory 63(1): 31---67, 2010), edge-colouring (Machado et al. in Theor Comput Sci 411(7---9): 1221---1234, 2010) and total-colouring (Machado and de Figueiredo in Discrete Appl Math 159(16): 1851---1864, 2011). Unichord-free graphs proved to have a rich structure that can be used to obtain interesting results with respect to the study of the complexity of colouring problems. In particular, several surprising complexity dichotomies of colouring problems are found in subclasses of unichord-free graphs. In the present work, we investigate clique-colouring and biclique-colouring problems restricted to unichord-free graphs. We show that the clique-chromatic number of a unichord-free graph is at most 3, and that the 2-clique-colourable unichord-free graphs are precisely those that are perfect. Moreover, we describe an O(nm)-time algorithm that returns an optimal clique-colouring of a unichord-free graph input. We prove that the biclique-chromatic number of a unichord-free graph is either equal to or one greater than the size of a largest twin set. Moreover, we describe an $$O(n^2m)$$O(n2m)-time algorithm that returns an optimal biclique-colouring of a unichord-free graph input. The clique-chromatic and the biclique-chromatic numbers are not monotone with respect to induced subgraphs. The biclique-chromatic number presents an extra unexpected difficulty, as it is not the maximum over the biconnected components, which we overcome by considering additionally the star-biclique-chromatic number.