  A connected graph is 2K2-free if it does not contain a pair of independent
edges as an induced subgraph. In this paper, we present the structural
characterization of minimal vertex separator and show that there are polynomial
number of minimal vertex separators in 2K2-free graphs. Further, using the
enumeration we show that finding minimum connected vertex separator in 2K2-free
graphs is polynomial time solvable. We highlight that finding minimum connected
vertex separator is NP-complete in Chordality 5 graphs, which is a super graph
class of 2K2-free graphs. Other study includes, enumeration of all distinct
maximal independent sets and testing 2K2-free graphs. Also, we present an
polynomial time algorithm for feedback vertex set problem in the subclass of
2K2-free graphs.
