Combinatorics and algorithms for augmenting graphs

TL;DR

This paper explores the concept of augmenting graphs, extending their application to solve the maximum independent set problem more efficiently, and identifies the minimal infinite classes of such graphs.

Contribution

It characterizes the minimal infinite classes of augmenting graphs and extends polynomial-time solvability of the MIS problem using this framework.

Findings

Identified minimal infinite classes of augmenting graphs.

Extended polynomial-time algorithms for MIS.

Enhanced understanding of augmenting graph structures.

Abstract

The notion of augmenting graphs generalizes Berge's idea of augmenting chains, which was used by Edmonds in his celebrated solution of the maximum matching problem. This problem is a special case of the more general maximum independent set (MIS) problem. Recently, the augmenting graph approach has been successfully applied to solve MIS in various other special cases. However, our knowledge of augmenting graphs is still very limited, and we do not even know what the minimal infinite classes of augmenting graphs are. In the present paper, we find an answer to this question and apply it to extend the area of polynomial-time solvability of the maximum independent set problem.