List $k$-Colouring $P_t$-Free Graphs: a Mim-width Perspective

A colouring of a graph $G=(V,E)$ is a mapping $c\colon V\to \{1,2,\ldots\}$ such that $c(u)\neq c(v)$ for every two adjacent vertices $u$ and $v$ of $G$. The {\sc List $k$-Colouring} problem is to decide whether a graph $G=(V,E)$ with a list $L(u)\subseteq \{1,\ldots,k\}$ for each $u\in V$ has a colouring $c$ such that $c(u)\in L(u)$ for every $u\in V$. 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 List $3$-Colouring is polynomial-time solvable for $(K_{1,s}^1,P_t)$-free graphs for every $t\geq 1$ and $s\geq 1$. We generalize their result to List $k$-Colouring for every $k\geq 1$. Our result also generalizes the known result that for every $k\geq 1$ and $s\geq 0$, 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 $s\geq 1$ 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 $k\geq 1$, $s\geq 1$, $t\geq 1$, 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.