We study the computational complexity of two well-known graph transversal problems, namely Subset Feedback Vertex Set and Subset Odd Cycle Transversal, by restricting the input to $H$-free graphs, that is, to graphs that do not contain some fixed graph~$H$ as an induced subgraph. By combining known and new results, we determine the computational complexity of both problems on $H$-free graphs for every graph $H$ except when $H=sP_1+P_4$ for some $s\geq 1$. As part of our approach, we introduce the Subset Vertex Cover problem and prove that it is polynomial-time solvable for $(sP_1+P_4)$-free graphs for every $s\geq 1$.
翻译:我们研究两个众所周知的图表横向问题的计算复杂性,即Subset Fertex Set 和 Subset Vertex Odd course Transversal, 其方法是将输入限制在不含$-free $的图表上, 也就是说, 限制在不含固定图形~$-H$的图表上, 作为诱导的子图。 通过将已知结果和新结果结合起来, 我们确定每个图表的不含$-H0的图表上这两个问题的计算复杂性, 除非当美元=s_ 1+P_4美元时, 某些美元=s\geq 1美元时。 作为我们的方法的一部分, 我们引入了 Subset Vertex 覆盖问题, 并证明每1美元(s_ 1+P_ 4) 的无$(sgeget 1美元) 的图形是可多音- 时间隔离的。