  In this paper, our goal is to characterize two graph classes based on the
properties of minimal vertex (edge) separators. We first present a structural
characterization of graphs in which every minimal vertex separator is a stable
set. We show that such graphs are precisely those in which the induced
subgraph, namely, a cycle with exactly one chord is forbidden. We also show
that deciding maximum such forbidden subgraph is NP-complete by establishing a
polynomial time reduction from maximum induced cycle problem [1]. This result
is of independent interest and can be used in other combinatorial problems.
Secondly, we prove that a graph has the following property: every minimal edge
separator induces a matching (that is no two edges share a vertex in common) if
and only if it is a tree.
