A graph is $k$-vertex-critical if $\chi(G)=k$ but $\chi(G-v)<k$ for all $v\in V(G)$ and $(G,H)$-free if it contains no induced subgraph isomorphic to $G$ or $H$. We show that there are only finitely many $k$-vertex-critical $(2P_2,H)$-free graphs for all $k$ when $H$ is isomorphic to any of the following graphs of order $5$: $bull$, $chair$, $claw+P_1$, or $\overline{diamond+P_1}$. The latter three are corollaries of more general results where $H$ is isomorphic to $(m, \ell)$-$squid$ for $m=3,4$ and any $\ell\ge 1$ where an $(m,\ell)$-$squid$ is the graph obtained from an $m$-cycle by attaching $\ell$ leaves to a single vertex of the cycle. For each of the graphs $H$ above and any fixed $k$, our results imply the existence of polynomial-time certifying algorithms for deciding the $k$-colourability problem for $(2P_2,H)$-free graphs. Further, our structural classifications allow us to exhaustively generate, with aid of computer search, all $k$-vertex-critical $(2P_2,H)$-free graphs for $k\le 7$ when $H=bull$ or $H=(4,1)$-$squid$ (also known as $banner$).

翻译：一个图是$k$-顶点临界图，若满足$\chi(G)=k$但对所有$v\in V(G)$有$\chi(G-v)<k$；若图中不含与$G$或$H$同构的诱导子图，则称其为$(G,H)$-free图。我们证明：当$H$同构于以下任意一个$5$阶图时，对于所有$k$，$k$-顶点临界$(2P_2,H)$-free图仅有有限多个：$bull$、$chair$、$claw+P_1$或$\overline{diamond+P_1}$。后三个结论是更一般结果的推论，其中$H$同构于$(m, \ell)$-$squid$（$m=3,4$且任意$\ell\ge 1$），而$(m,\ell)$-$squid$定义为从$m$元环的一个顶点上附加$\ell$个叶子节点所得到的图。对于上述每个$H$和任意固定的$k$，我们的结果保证了存在多项式时间可验证的算法，用于判定$(2P_2,H)$-free图的$k$-可着色问题。此外，通过结构分类，我们借助计算机搜索穷举生成了所有$k\le 7$时的$k$-顶点临界$(2P_2,H)$-free图，其中$H=bull$或$H=(4,1)$-$squid$（也称为$banner$）。