Generalized Edmonds-Sterboul-Deming configurations. Part 1: Sterboul-Deming graphs

TL;DR

This paper introduces generalized graph configurations called Jflowers and Jposies, extending classical concepts to better characterize non-König-Egerváry graphs and unify various matching configurations.

Contribution

It defines new generalized configurations and proves their equivalence with classical ones, providing a unified framework for analyzing Sterboul-Deming graphs.

Findings

Sets of vertices covered by classical and generalized configurations coincide.

Unified characterization of Sterboul-Deming graphs.

Enhanced tools for studying maximum matchings.

Abstract

We introduce two new types of graph configurations, the Jflower and the Jposy, which generalize the classical flower and posy configurations of Edmonds, Sterboul, and Deming in the context of maximum matchings. These generalized configurations allow greater flexibility in characterizing non-Konig-Egerv\'ary graphs and provide new tools for studying matching-theoretic properties. Our main result shows that the sets of vertices covered by classical configurations (flowers and posies), restricted configurations (Tposies), and generalized configurations (Jflowers and Jposies) coincide. This equivalence yields a unified characterization of what we call Sterboul-Deming graphs, graphs in which every vertex belongs to some configuration relative to an appropriate maximum matching.