Graphs in which every minimal separator is an independent set

Question

Background: Let $u, v$ be two vertices of an undirected graph $G=(V,E)$. A vertex set $S\subseteq V$ is a $u,v$-separator if $u$ and $v$ belong to different connected components of $G-S$. If no proper subset of a $u,v$-separator $S$ is a $u,v$-separator then $S$ is a minimal $u,v$-separator. A vertex set $S\subseteq V$ is a (minimal) separator if there exist vertices $u, v$ such that $S$ is a (minimal) $u,v$-separator.

A well-known theorem of G. Dirac states that a graph has no induced cycles of length at least four (called triangulated or chordal graph) if and only if every of its minimal separators is a clique. It is also well-known that triangulated graphs can be recognized in polynomial time.

My questions: What are graphs in which every minimal separator is an independent set? Are these graphs studied? And what is the recognition complexity of these graphs? Examples for such graphs include trees and cycles.

Answer 1

Your graphs have been characterized by this paper http://arxiv.org/pdf/1103.2913.pdf.

Edit: In the paper above it is proved that graphs in which every minimal separator is an independent set are exactly those containing no cycle with exactly one chord.

Graphs containing no cycle with exactly one chord have been studied in depth by Trotignon and Vuskovic, A Structure Theorem for Graphs with No Cycle with a Unique Chord and Its Consequences, J. Graph Theory 63 (2010) 31-67 DOI. As a result of this paper, these graphs can be recognized in polynomial time. (However, this paper did not point out the connection to independent minimal separators!)