Sterboul-Deming Graphs: Characterizations

TL;DR

This paper characterizes Sterboul--Deming graphs, a class related to K"onig--Egerváry graphs, by providing structural decompositions, algorithms, and broad inclusion criteria, linking classical graph theory concepts.

Contribution

It introduces several new characterizations and decomposition algorithms for Sterboul--Deming graphs, expanding understanding of their structure and relation to other graph classes.

Findings

Provides a constructive algorithm for graphs with perfect matchings.

Extends analysis to graphs with unique perfect matchings using Gallai--Edmonds decomposition.

Shows the class includes all graphs with an odd cycle factor.

Abstract

A graph is said to be a Sterboul--Deming graph if $KE(G)=\emptyset$, that is, if every vertex of $G$ belongs to a posy or a flower (structures introduced by Sterboul, Deming, and Edmonds). These graphs can be regarded as the structural counterparts of K\"onig--Egerv\'ary graphs. In this paper, we present several characterizations of Sterboul--Deming graphs. We first study the case of graphs with a perfect matching and with a unique perfect matching, providing a constructive algorithm to obtain the decomposition $(SD(G), KE(G))$. Then, we extend the analysis to the general case through the Gallai--Edmonds decomposition. In addition, we show that the class of Sterboul--Deming graphs is remarkably broad: it contains all graphs having a $\{C_n : n \textnormal{ odd}\}$-factor, providing a simple structural criterion for identifying such graphs. These results establish new connections between classical decomposition theorems and the internal structure of non--K\"onig--Egerv\'ary graphs.