Vertex-critical graphs in subfamilies of $(P_4+\ell P_1)$-free graphs

A graph $G$ is $k$-vertex-critical if $χ(G)=k$ but $χ(G-v)<k$ for all $v\in V(G)$. In this paper we make progress on the open problem of the finiteness of $k$-vertex-critical $(P_4+\ell P_1)$-free graphs by showing that there are only finitely many $k$-vertex-critical graphs in the following subfamilies of $(P_4+\ell P_1)$-free graphs for all $k\ge 1$ and $\ell\ge 0$: $\bullet$ $(P_4+\ell P_1,\text{chair})$-free graphs, $\bullet$ $(P_4+\ell P_1,P_5,\text{bull})$-free graphs, and $\bullet$ $(P_4+\ell P_1,P_5,\text{cricket})$-free graphs. In fact, all but the first of these are special cases of our general result that there are only finitely many $k$-vertex-critical $(P_4+\ell P_1,B_{4}(m),B_{3}(m)^{+})$-free graphs for all $k\ge 1$ and $\ell,m\ge 0$. Here $B_{n}(m)$ is the graph obtained from a path of order $n$ by identifying one of its leaves with the centre vertex of $K_{1,m}$ and $B_{n}(m)^{+}$ is the graph obtained by identifying an edge of $K_3$ with the edge of $B_{n}(m)$ with endpoints of degrees $2$ and $m$, respectively. Our results imply the existence of simple polynomial-time certifying algorithms to decide the $k$-colourability of all graphs in these subfamilies for every fixed $k$. We also show that $χ(G)\le \ell+2$ for all $(P_4+\ell P_1,K_3)$-free graphs and all $\ell\ge 0$, improving the previously known upper bound of $2\ell+2$ that followed from Randerath and Schiermeyer's 2004 result on $(P_t,K_3)$-free graphs. More generally, we provide a $χ$-bound in $O(\ell^{ω-1})$ for $(P_4+\ell P_1)$-free graphs which improves the bound of $(2\ell+2)^{ω-1}$ which followed from Gravier, Hoàng and Maffray in 2003 for $P_{t}$-free graphs.