We prove new complexity results for Feedback Vertex Set and Even Cycle Transversal on $H$-free graphs, that is, graphs that do not contain some fixed graph $H$ as an induced subgraph. In particular, we prove that both problems are polynomial-time solvable for $sP_3$-free graphs for every integer $s\geq 1$. Our results show that both problems exhibit the same behaviour on $H$-free graphs (subject to some open cases). This is in part explained by a new general algorithm we design for finding in a graph $G$ a largest induced subgraph whose blocks belong to some finite class ${\cal C}$ of graphs. We also compare our results with the state-of-the-art results for the Odd Cycle Transversal problem, which is known to behave differently on $H$-free graphs.
翻译：在无H$的图表上,我们为“Vertex Set”和“Even Cycle Transversal”的反馈“Vertex Set”和“Even Centural Transversal”提供了新的复杂结果,也就是说,图表中并不包含固定的图形$H$,作为诱导的子图。特别是,我们证明,对于每个整数$s\geq 1美元,这两个问题都是多米时间的无美元图表。我们的结果显示,这两个问题在无H$的图表上都表现了相同的行为(但有些尚未解决的案例除外 ) 。 这部分是由于我们设计了一种新的通用算法, 以在图表中找到一个最大的“$G$”的引出子图, 其块属于某个限定级的$=C}图表。 我们还比较了我们的结果与“奇周期” Transversal 问题的最新结果, 这个问题在无H$H$美元图表上表现不同。
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