Extremal minimal bipartite matching covered graphs

2024-04-09

arXiv.org

AI Summary

This paper characterizes the five extremal classes of minimally bipartite matching-covered graphs—i.e., all graphs attaining the lower bounds under five classical graph parameters, such as having exactly $2(m - n + 2)$ vertices of degree two. Methodologically, it introduces a unified “double-leaf-corresponding gluing” construction paradigm, grounded in ear decompositions, the Lovász–Plummer structural theory of matchings, and standard matching transformations. The approach yields the first complete characterization of all five extremal families: three are explicitly constructed, while the remaining two are reduced to known classes via established equivalences. The results significantly advance the extremal theory of bipartite matching-covered graphs, clarify intrinsic relationships among the four extremal families originally conjectured by Lovász and Plummer, and provide a strengthened foundation for conjectures on the boundary of $k$-extendability in bipartite graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

A connected graph, on four or more vertices, is matching covered if every edge is present in some perfect matching. An ear decomposition theorem (similar to the one for $2$-connected graphs) exists for bipartite matching covered graphs due to Hetyei. From the results and proofs of Lov'asz and Plummer, that rely on Hetyei's theorem, one may deduce that any minimal bipartite matching covered graph has at least $2(m-n+2)$ vertices of degree two (where minimal means that deleting any edge results in a graph that is not matching covered); such a graph is said to be extremal if it attains the stated lower bound. In this paper, we provide a complete characterization of the class of extremal minimal bipartite matching covered graphs. In particular, we prove that every such graph $G$ is obtained from two copies of a tree devoid of degree two vertices, say $T$ and $T'$, by adding edges -- each of which joins a leaf of $T$ with the corresponding leaf of $T'$. Apart from the aforementioned bound, there are four other bounds that appear in, or may be deduced from, the work of Lov'asz and Plummer. Each of these bounds leads to a notion of extremality. In this paper, we obtain a complete characterization of all of these extremal classes and also establish relationships between them. Two of our characterizations are in the same spirit as the one stated above. For the remaining two extremal classes, we reduce each of them to one of the already characterized extremal classes using standard matching theoretic operations.

Problem

• Characterize extremal minimal bipartite matching covered graphs
• Establish relationships between different extremal graph classes
• Investigate bounds for minimal k-extendable bipartite graphs

Innovation

• Characterizes extremal minimal bipartite matching covered graphs
• Uses tree copies and leaf connections
• Reduces extremal classes via matching operations