List k-Colouring P_t-Free Graphs with No Induced 1-Subdivision of K_1,s: a Mim-width Perspective
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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 proved that List 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. We show our result by proving 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".
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