For a family of graphs $\mathcal{F}$, Weighted $\mathcal{F}$-Deletion is the problem for which the input is a vertex weighted graph $G=(V,E)$ and the goal is to delete $S\subseteq V$ with minimum weight such that $G\setminus S\in\mathcal{F}$. Designing a constant-factor approximation algorithm for large subclasses of perfect graphs has been an interesting research direction. Block graphs, 3-leaf power graphs, and interval graphs are known to admit constant-factor approximation algorithms, but the question is open for chordal graphs and distance-hereditary graphs. In this paper, we add one more class to this list by presenting a constant-factor approximation algorithm when $F$ is the intersection of chordal graphs and distance-hereditary graphs. They are known as ptolemaic graphs and form a superset of both block graphs and 3-leaf power graphs above. Our proof presents new properties and algorithmic results on inter-clique digraphs as well as an approximation algorithm for a variant of Feedback Vertex Set that exploits this relationship (named Feedback Vertex Set with Precedence Constraints), each of which may be of independent interest.

中文翻译：

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弦/距离遗传顶点删除的常数因子近似

对于图族 $\mathcal{F}$，Weighted $\mathcal{F}$-Deletion 是输入为顶点加权图 $G=(V,E)$ 并且目标是删除的问题$S\subseteq V$ 具有最小权重，使得 $G\setminus S\in\mathcal{F}$。为完美图的大子类设计常数因子近似算法一直是一个有趣的研究方向。已知块图、3 叶幂图和区间图允许使用常数因子近似算法，但问题对于弦图和距离遗传图是开放的。在本文中，当 $F$ 是弦图和距离遗传图的交集时，我们通过提出一种常数因子近似算法向该列表中添加了一个类。它们被称为托勒密图，形成上面的块图和 3 叶幂图的超集。我们的证明提出了群间有向图的新属性和算法结果，以及利用这种关系的反馈顶点集变体的近似算法（称为具有优先约束的反馈顶点集），每个可能是独立的兴趣。