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

Daniel A. Jaume, Cristian Panelo, Kevin Pereyra

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-Egerváry 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.

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

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-Egerváry 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.
Paper Structure (7 sections, 12 theorems, 7 equations, 13 figures)

This paper contains 7 sections, 12 theorems, 7 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Jflower
  4. Jposies
  5. Jposies mark the same vertices as posies
  6. Jflower vertices are Jposy vertices or flower vertices
  7. Main result

Key Result

Lemma 4.1

[lemma]lem:JObs Let $G$ be a graph and $M$ be a maximum matching of $G$. If $H$ is an $M$-Tposy of $G$ and $v \in V(H)$, then there exists a non-trivial $M$-$mm$-alternating path in $H$ from $v$ to the base of one the blossoms of $H$.

Figures (13)

  • Figure 1: Edmonds configurations
  • Figure 2: $M$-Jflower
  • Figure 3: Examples of posy configurations
  • Figure 4: Case 1: Three ears: $2P3=2,8,9,3$, $3P4=3,10,11,4$ and $3P5=3,12,13,5$.
  • Figure 5: Case 2: Three ears: $2P3=2,17,16,3$, $9P10=9,14,15,10$ and $6P8=6,18,19,8$.
  • ...and 8 more figures

Theorems & Definitions (20)

  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • proof
  • Theorem 5.1: Sterboul's Characterization of Kőnig-Egerváry graphs sterboul1979characterization
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • proof
  • Theorem 5.4: bonomo2013forbidden
  • ...and 10 more