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Constructive Characterization of Critical Bipartite Grafts

Nanao Kita

Abstract

Factor-critical graphs are a classical concept in matching theory that constitute an important component of the Gallai-Edmonds canonical decomposition and Edmonds' algorithm for maximum matchings. Lovász provided a constructive characterization of factor-critical graphs in terms of ear decompositions. This characterization has been a useful inductive tool for studying factor-critical graphs and also connects them with Edmonds' algorithm. Joins in grafts, also known as $T$-joins in graphs, are a classical variant of matchings proposed in terms of parity. Minimum joins and grafts are generalizations of perfect matchings and graphs with perfect matchings, respectively. Accordingly, graft analogues of fundamental concepts and results from matching theory, such as canonical decompositions, will develop the theory of minimum join. In this paper, we propose a new concept, critical quasicombs, as a bipartite graft analogue of factor-critical graphs and provide a constructive characterization of critical quasicombs using a graft version of ear decompositions. This characterization can be considered as a bipartite graft analogue of Lovász' result. From our results, the Dulmage-Mendelsohn canonical decomposition, originally a theory for bipartite graphs, has been generalized for bipartite grafts.

Constructive Characterization of Critical Bipartite Grafts

Abstract

Factor-critical graphs are a classical concept in matching theory that constitute an important component of the Gallai-Edmonds canonical decomposition and Edmonds' algorithm for maximum matchings. Lovász provided a constructive characterization of factor-critical graphs in terms of ear decompositions. This characterization has been a useful inductive tool for studying factor-critical graphs and also connects them with Edmonds' algorithm. Joins in grafts, also known as -joins in graphs, are a classical variant of matchings proposed in terms of parity. Minimum joins and grafts are generalizations of perfect matchings and graphs with perfect matchings, respectively. Accordingly, graft analogues of fundamental concepts and results from matching theory, such as canonical decompositions, will develop the theory of minimum join. In this paper, we propose a new concept, critical quasicombs, as a bipartite graft analogue of factor-critical graphs and provide a constructive characterization of critical quasicombs using a graft version of ear decompositions. This characterization can be considered as a bipartite graft analogue of Lovász' result. From our results, the Dulmage-Mendelsohn canonical decomposition, originally a theory for bipartite graphs, has been generalized for bipartite grafts.

Paper Structure

This paper contains 30 sections, 19 theorems, 6 figures.

Table of Contents

  1. Introduction
  2. Joins and Grafts
  3. Toward Canonical Decomposition Theory for Joins in Grafts
  4. Canonical Decompositions in Matching Theory
  5. Factor-Components and Canonical Decompositions for Factorizable Graphs
  6. Extension to Grafts
  7. Factor-Critical Graphs
  8. Basics
  9. Lovász' Constructive Characterization of Factor-Critical Graphs
  10. Factor-Critical Graphs and Cathedral Decomposition
  11. Our Aim and Results
  12. Consequence of Our Results
  13. Organization of This Paper
  14. Definitions
  15. Basic Notation
  16. ...and 15 more sections

Key Result

Proposition 4.2

Let $(G, T)$ be a graft, and let $F$ be a minimum join. Then, $\lambda(u, v; F; G, T) = \nu(G, T\Delta \{u, v\}) - \nu(G, T)$ for every $u, v\in V(G)$.

Figures (6)

  • Figure 1: Relationships among factor-critical graph, factor-critical grafts, and critical quasicombs.
  • Figure 2: Critical quasicomb $(G, T; A, B)$ with root $r$ and one of its minimum joins $F$: The gray and white points indicate vertices in $T$ and $V(G)\setminus T$, respectively, and the upper and lower vertices form color classes $B$ and $A$, respectively. Bold lines indicate edges in $F$.
  • Figure 3: Graft $(G_1, T_1; A_1, B_1)$ and ear graft $(P_1, T_1'; A_1', B_1')$.
  • Figure 4: Graft $(G_2, T_2; A_2, B_2)$ and ear graft $(P_2, T_2'; A_2', B_2')$.
  • Figure 5: Graft $(G_3, T_3; A_3, B_3)$ and ear graft $(P_3, T_3'; A_3', B_3')$.
  • ...and 1 more figures

Theorems & Definitions (48)

  • Definition 4.1
  • Proposition 4.2: Sebö DBLP:journals/jct/Sebo90
  • Lemma 4.3: see also Sebö DBLP:journals/jct/Sebo90
  • Definition 5.1
  • Theorem 5.2: Lovász lovasz1972notelp1986
  • Remark 5.3
  • Proposition 5.4: see Lovász and Plummer lp1986
  • Definition 6.1
  • Proposition 6.2: Sebö DBLP:journals/jct/Sebo90
  • Definition 6.4
  • ...and 38 more