Chordal graphs are characterized as the intersection graphs of subtrees in a tree and such a representation is known as the tree model. Restricting the characterization results in well-known subclasses of chordal graphs such as interval graphs or split graphs. A typical example that behaves computationally different in subclasses of chordal graph is the \textsc{Subset Feedback Vertex Set} (SFVS) problem: given a graph $G=(V,E)$ and a set $S\subseteq V$, SFVS asks for a minimum set of vertices that intersects all cycles containing a vertex of $S$. SFVS is known to be polynomial-time solvable on interval graphs, whereas SFVS remains \NP-complete on split graphs and, consequently, on chordal graphs. Towards a better understanding of the complexity of SFVS on subclasses of chordal graphs, we exploit structural properties of a tree model in order to cope with the hardness of SFVS. Here we consider variants of the \emph{leafage} that measures the minimum number of leaves in a tree model. We show that SFVS can be solved in polynomial time for every chordal graph with bounded leafage. In particular, given a chordal graph on $n$ vertices with leafage $\ell$, we provide an algorithm for SFVS with running time $n^{O(\ell)}$. Pushing further our positive result, it is natural to consider a slight generalization of leafage, the \emph{vertex leafage}, which measures the smallest number among the maximum number of leaves of all subtrees in a tree model. However, we show that it is unlikely to obtain a similar result, as we prove that SFVS remains \NP-complete on undirected path graphs, i.e., graphs having vertex leafage at most two. Moreover, we strengthen previously-known polynomial-time algorithm for SFVS on directed path graphs that form a proper subclass of undirected path graphs and graphs of mim-width one.

中文翻译：

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通过叶子计算子集反馈顶点集

弦图的特征是树中子树的相交图，这种表示称为树模型。限制特征化会导致形成弦图的众所周知的子类，例如间隔图或分裂图。\ textsc {子集反馈顶点集}（SFVS）问题是在弦图子类中计算上不同的一个典型示例：给定一个图$ G =（V，E）$和一个集合$ S \ subseteq V $，SFVS要求与包含$ S $顶点的所有循环相交的最小顶点集。已知SFVS在间隔图上是多项式时间可解的，而SFVS在分裂图上以及因此在弦图上仍保持\ NP-complete。为了更好地理解弦图的子类上SFVS的复杂性，我们利用树模型的结构特性来应对SFVS的硬度。在这里，我们考虑\ emph {leafage}的变体，该变体测量树模型中的最小叶子数。我们表明，SFVS可以在多项式时间内对每个有界叶子的弦图进行求解。特别地，给定具有叶子$ \ ell $的$ n $顶点的弦图，我们为运行时间为$ n ^ {O（\ ell）} $的SFVS提供了一种算法。进一步推动我们的积极结果，自然会考虑对叶子的略微概括，即\ emph {vertex leafage}，它测量树模型中所有子树的最大叶子数中的最小数。但是，我们证明不太可能获得相似的结果，因为我们证明SFVS在无向路径图（即具有最多两个顶点叶子的图）上保持\ NP完全。而且，