  The family of graphs that can be constructed from isolated vertices by
disjoint union and graph join operations are called cographs. These graphs can
be represented in a tree-like representation termed parse tree or cotree. In
this paper, we study some popular combinatorial problems restricted to
cographs. We first present a structural characterization of minimal vertex
separators in cographs. Further, we show that listing all minimal vertex
separators and the complexity of some constrained vertex separators are
polynomial-time solvable in cographs. We propose polynomial-time algorithms for
connectivity augmentation problems and its variants in cographs, preserving the
cograph property. Finally, using the dynamic programming paradigm, we present a
generic framework to solve classical optimization problems such as the longest
path, the Steiner path and the minimum leaf spanning tree problems restricted
to cographs, our framework yields polynomial-time algorithms for all three
problems.
