Towards Constant-Factor Approximation for Chordal/Distance-Hereditary Vertex Deletion

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 $$\mathcal {F}$$ 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.