A typical example that behaves computationally different in subclasses of chordal graphs is the \textsc{Subset Feedback Vertex Set} (SFVS) problem: given a vertex-weighted graph $G=(V,E)$ and a set $S\subseteq V$, the \textsc{Subset Feedback Vertex Set} (SFVS) problem asks for a vertex set of minimum weight 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 $n^{O(\ell)}$. We complement our result by showing that SFVS is \W[1]-hard parameterized by $\ell$. 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 rooted path graphs that form a proper subclass of undirected path graphs and graphs of mim-width one.
翻译:在chordal 图形的子类中,一个在计算时出现不同表现的典型示例是:\ textsc{Subset反馈 VertexSet} (SFVS) 问题:鉴于一个顶点加权图形$G=(V,E) 和一套美元Ssubseteq V$,\ textsc{Subs反馈VdexSet} (SFVS) 问题需要一套最低重量的顶点组,它交叉包含一个$的顶点。SFVS是已知的极点,而SFVS在间图中是多时的,而SFServiS在分点上仍然完整。我们利用一个树模型的结构特性来应付SFVS的硬度。我们在这里考虑的是所有变数。SFOVSlextreal drealdalal dismaldality,我们通过每部位模型来测量其最低叶数。我们用一个固定的货币值显示一个固定的SFSFSF,我们用一个固定的阵点显示一个固定的阵点显示。