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.