Sterboul-Deming Graphs: Characterizations

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 König--Egerváry 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--König--Egerváry graphs.

Figures (5)

• Figure 1: Illustration of Algorithm 1
• Figure 2: Illustration of the reduced form $\mathcal{R}(G)$ of $G$
• Figure 3: Illustration of the proof of \ref{['ilustr12a']}
• Figure 4: Example of a graph $G$ with a $\{C_{n}:n\textnormal{ odd}\}$-factor. Note that $G$ is a Sterboul--Deming graph. This is an example of \ref{['thm410ll']}.
• Figure 5: Illustration of \ref{['corolarioasdarriba2']}

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2: jaume2025confpart1
• Theorem 1.3
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• Theorem 3.4: agusvalenota
• Theorem 3.5