List k-Colouring P_t-Free Graphs: a Mim-width Perspective [article]

Nick Brettell, Jake Horsfield, Andrea Munaro, Daniel Paulusma
2021 arXiv   pre-print
A colouring of a graph G=(V,E) is a mapping c V→{1,2,...} such that c(u)≠ c(v) for every two adjacent vertices u and v of G. The List k-Colouring problem is to decide whether a graph G=(V,E) with a list L(u)⊆{1,...,k} for each u∈ V has a colouring c such that c(u)∈ L(u) for every u∈ V. Let P_t be the path on t vertices and let K_1,s^1 be the graph obtained from the (s+1)-vertex star K_1,s by subdividing each of its edges exactly once.Recently, Chudnovsky, Spirkl and Zhong (DM 2020) proved that
more » ... ist 3-Colouring is polynomial-time solvable for (K_1,s^1,P_t)-free graphs for every t≥ 1 and s≥ 1. We generalize their result to List k-Colouring for every k≥ 1. Our result also generalizes the known result that for every k≥ 1 and s≥ 0, List k-Colouring is polynomial-time solvable for (sP_1+P_5)-free graphs, which was proven for s=0 by Hoàng, Kamiński, Lozin, Sawada, and Shu (Algorithmica 2010) and for every s≥ 1 by Couturier, Golovach, Kratsch and Paulusma (Algorithmica 2015). We show our result by proving boundedness of an underlying width parameter. Namely, we show that for every k≥ 1, s≥ 1, t≥ 1, the class of (K_k,K_1,s^1,P_t)-free graphs has bounded mim-width and that a corresponding branch decomposition is "quickly computable" for these graphs.
arXiv:2008.01590v2 fatcat:vkjvicp64bfvvlu2fssul32swi