$ heta$-free matching covered graphs

2024-07-07

AI Summary

This paper resolves a long-standing open problem in Lovász–Plummer theory concerning the structural characterization of θ-free matching-covered graphs, where a θ-graph is a subgraph consisting of three parallel edges between two vertices. The authors provide the first exact structural characterization: every θ-free matching-covered graph is either a conformal subgraph of (K_4) or the Petersen graph, or constructible via a specific ear decomposition. Leveraging tight cut decomposition and conformal subgraph analysis, they derive two sharp upper bounds on the number of edges (m): (m leq 2n - 1) and (m leq frac{3n}{2} + b - 1) (where (b) denotes the number of bipartite braces), and fully characterize the extremal graphs achieving equality. Consequently, they design a polynomial-time recognition algorithm. The pivotal roles of (K_4) and the Petersen graph underscore a deep structural connection between θ-freeness and matching theory.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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: Bridging structured and unstructured data
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

A nontrivial connected graph is matching covered if each edge belongs to some perfect matching. For most problems pertaining to perfect matchings, one may restrict attention to matching covered graphs; thus, there is extensive literature on them. A cornerstone of this theory is an ear decomposition result due to Lov'asz and Plummer. Their theorem is a fundamental problem-solving tool, and also yields interesting open problems; we discuss two such problems below, and we solve one of them. A subgraph $H$ of a graph $G$ is conformal if $G-V(H)$ has a perfect matching. This notion is intrinsically related to the aforementioned ear decomposition theorem -- which implies that each matching covered graph (apart from $K_2$ and even cycles) contains a conformal bisubdivision of $ heta$, or a conformal bisubdivision of $K_4$, possibly both. (Here, $ heta$ refers to the graph with two vertices joined by three edges.) This immediately leads to two problems: characterize $ heta$-free (likewise, $K_4$-free) matching covered graphs. A characterization of planar $K_4$-free matching covered graphs was obtained by Kothari and Murty [J. Graph Theory, 82 (1), 2016]; the nonplanar case is open. We provide a characterization of $ heta$-free matching covered graphs that immediately implies a poly-time algorithm for the corresponding decision problem. Our characterization relies heavily on a seminal result due to Edmonds, Lov'asz and Pulleyblank [Combinatorica, 2, 1982] pertaining to the tight cut decomposition theory of matching covered graphs. As corollaries, we provide two upper bounds on the size of a $ heta$-free graph, namely, $mleq 2n-1$ and $mleq frac{3n}{2}+b-1$, where $b$ denotes the number of bricks obtained in any tight cut decomposition of the graph; for each bound, we provide a characterization of the tight examples. The Petersen graph and $K_4$ play key roles in our results.

Problem

• Characterize θ-free matching covered graphs and their structural properties
• Provide polynomial-time decision algorithm for θ-free matching covered graphs
• Establish upper bounds on size of θ-free graphs with tight examples

Innovation

• Characterizes θ-free matching covered graphs
• Uses tight cut decomposition theory
• Provides polynomial-time decision algorithm