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We prove that the class of $(K_t,sP_1+P_5)$-free graphs has bounded mim-widthfor every $s\geq 0$ and $t\geq 1$, and that there is a polynomial-timealgorithm that, given a graph in the class, computes a branch decomposition ofconstant mim-width. A large number of \NP-complete graph problems becomepolynomial-time solvable on graph classes with bounded mim-width and for whicha branch decomposition is quickly computable. The $k$-Colouring problem is anexample of such a problem. For this problem, we may assume that the input graphis $K_{k+1}$-free. Then, as a consequence of our result, we obtain a new prooffor 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. Infact, our findings show that the underlying reason for this polynomial-timealgorithm is that the class has bounded mim-width.

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