Colouring (sP_1+P_5)-Free Graphs: a Mim-Width Perspective
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We prove that the class of (K_t,sP_1+P_5)-free graphs has bounded mim-width for every s≥ 0 and t≥ 1, and that there is a polynomial-time algorithm that, given a graph in the class, computes a branch decomposition of constant mim-width. A large number of -complete graph problems become polynomial-time solvable on graph classes with bounded mim-width and for which a branch decomposition is quickly computable. The k-Colouring problem is an example of such a problem. For this problem, we may assume that the input graph is K_k+1-free. Then, as a consequence of our result, we obtain a new proof for the known result that for every fixed k≥ 1 and s≥ 0, k-Colouring is polynomial-time solvable for (sP_1+P_5)-free graphs. In fact, our findings show that the underlying reason for this polynomial-time algorithm is that the class has bounded mim-width.
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