Egerv\'{a}ry graphs: Deming decompositions and independence structure

TL;DR

This paper presents a novel decomposition method for matchable graphs based on Deming's algorithm, revealing detailed independence and matching structures, and advancing understanding of $ ext{α}$-critical graphs.

Contribution

It introduces a new decomposition of matchable graphs into specific subgraphs with known independence and matching properties, refining previous structural results.

Findings

Decomposition into blossom pairs, $K_4$, or KE graphs with perfect matchings

Independence number is one less than matching number for blossom and $K_4$ subgraphs

Independence number equals matching number for KE subgraphs

Abstract

We leverage an algorithm of Deming [R.W. Deming, Independence numbers of graphs -- an extension of the Koenig-Egervary theorem, Discrete Math., 27(1979), no. 1, 23--33; MR534950] to decompose a matchable graph into subgraphs with a precise structure: they are either spanning even subdivisions of blossom pairs, spanning even subdivisions of the complete graph $K_4$, or a K\H{o}nig-Egerv\'{a}ry graph. In each case, the subgraphs have perfect matchings; in the first two cases, their independence numbers are one less than their matching numbers, while the independence number of the KE subgraph equals its matching number. This decomposition refines previous results about the independence structure of an arbitrary graph and leads to new results about $\alpha$-critical graphs.