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For $k \geq 3$, we prove (i) there is a finite number of $k$-vertex-critical$(P_2+\ell P_1)$-free graphs and (ii) $k$-vertex-critical $(P_3+P_1)$-freegraphs have at most $2k-1$ vertices. Together with previous research, theseresults imply the following characterization where $H$ is a graph of orderfour: There is a finite number of $k$-vertex-critical $H$-free graphs for fixed$k \geq 5$ if and only if $H$ is one of $\overline{K_4}, P_4, P_2 + 2P_1$, or$P_3 + P_1$. Our results imply the existence of new polynomial-time certifyingalgorithms for deciding the $k$-colorability of $(P_2+\ell P_1)$-free graphsfor fixed $k$.

Dichotomizing $k$-vertex-critical $H$-free graphs for $H$ of order four