  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.
