On 2K2-free graphs - Structural and Combinatorial View

TL;DR

This paper explores the structural properties of 2K2-free graphs, demonstrating polynomial-time algorithms for key problems like minimum connected vertex separator and feedback vertex set, and contrasting these with NP-complete cases in related graph classes.

Contribution

It provides a structural characterization of minimal vertex separators in 2K2-free graphs and develops polynomial algorithms for several problems within this class.

Findings

Polynomial number of minimal vertex separators in 2K2-free graphs

Polynomial-time algorithm for minimum connected vertex separator in 2K2-free graphs

NP-completeness of the problem in chordality 5 graphs

Abstract

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.