论文标题
二分法$ k $ -vertex-关键$ h $ - 订单$ h $四
Dichotomizing $k$-vertex-critical $H$-free graphs for $H$ of order four
论文作者
论文摘要
对于$ k \ geq 3 $，我们证明（i）有限数量的$ k $ - vertex-关键$（p_2+\ ell p_1）$ - 免费图形和（ii）$ k $ -vertex-Contrical $（p_3+p_1）$ - 最多的2k-1 $ vertices。与先前的研究一起，这些结果意味着以下表征，其中$ h $是第四订单的图：$ k $ -vertex-Contrical-tigartial $ h $ h $ - 固定$ k \ geq 5 $ if的无用图，并且仅当$ h $是$ h $是$ \ hub是$ \ edmepline {k_4}，p_4，p_2 + 2p + 2pp_1 $ $ p_1 $ p_1 $ p_1 $ p_1 $ p_1我们的结果意味着存在新的多项式时间认证算法，以确定$ K $ - 可溶性的$（p_2+\ ell p_1）$ - 固定$ k $的免费图。
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 $(P_3+P_1)$-free graphs have at most $2k-1$ vertices. Together with previous research, these results imply the following characterization where $H$ is a graph of order four: 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 certifying algorithms for deciding the $k$-colorability of $(P_2+\ell P_1)$-free graphs for fixed $k$.
