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.
翻译:弦图被描述为树中的子树的交叉图, 这样的表示方式被称为树型模型。 限制描述性能导致著名的chordal 图形的小类, 如间距图或分裂图。 一个典型的例子, 在chordal 图形的小类中, 运行计算方式不同 :\ textsc{ Subseption Vertex Set} (SFVS) 问题 : 给一个 $G=( V, E) 和一套 $S\subsetredial V$, SFVS 需要一套最小的垂直图。 SFVVS 将所有周期中包含 $S. SFVS 的顶端图解 。 SFS 的顶端点是 SFS 的底端点, SFS 的底点是 。 SFS 最深的底点是, SFS 的底点是 的底点是 。