Colouring $(sP_1+P_5)$-Free Graphs: a Mim-Width Perspective

We prove that the class of $(K_t,sP_1+P_5)$-free graphs has bounded mim-width for every $s\geq 0$ and $t\geq 1$, 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 \NP-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 $k\geq 1$ and $s\geq 0$, $k$-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.