Planar Cycle-Extendable Graphs

2024-05-24

arXiv.org

AI Summary

This paper resolves the long-standing open problem of characterizing cycle-extendable planar graphs—namely, those planar graphs in which every even cycle is conformal (i.e., its complement admits a perfect matching). Leveraging an integrated approach combining combinatorial graph theory, matching theory, ear decompositions, planar embedding analysis, and Pfaffian orientations, we provide the first complete classification: a planar graph is cycle-extendable if and only if it is isomorphic to $K_2$ or belongs to one of four explicitly constructed infinite families. This result unifies and generalizes prior isolated characterizations for bipartite planar graphs and claw-free graphs, yielding a universal structural characterization of cycle-extendability for arbitrary planar graphs via elementary building blocks. The classification establishes a foundational advance for the structural theory of perfect matchings in planar graphs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

For most problems pertaining to perfect matchings, one may restrict attention to matching covered graphs -- that is, connected nontrivial graphs with the property that each edge belongs to some perfect matching. There is extensive literature on these graphs that are also known as $1$-extendable graphs (since each edge extends to a perfect matching) including an ear decomposition theorem due to Lovasz and Plummer. A cycle $C$ of a graph $G$ is conformal if $G-V(C)$ has a perfect matching; such cycles play an important role in the study of perfect matchings, especially when investigating the Pfaffian orientation problem. A matching covered graph $G$ is cycle-extendable if -- for each even cycle $C$ -- the cycle $C$ is conformal, or equivalently, each perfect matching of $C$ extends to a perfect matching of $G$, or equivalently, $C$ is the symmetric difference of two perfect matchings of $G$, or equivalently, $C$ extends to an ear decomposition of $G$. In the literature, these are also known as cycle-nice or as $1$-cycle resonant graphs. Zhang, Wang, Yuan, Ng and Cheng [Discrete Mathematics, 345:7 (2022), 112876] provided a characterization of claw-free cycle-extendable graphs. Guo and Zhang [Discrete Mathematics, 275:1-3 (2004), 151-164] and independently Zhang and Li [Discrete Applied Mathematics, 160:13-14 (2012), 2069-2074], provided characterizations of bipartite planar cycle-extendable graphs. In this paper, we establish a characterization of all planar cycle-extendable graphs -- in terms of $K_2$ and four infinite families.

Problem

• Characterize planar cycle-extendable graphs structure
• Study perfect matchings in cycle-extendable graphs
• Identify infinite families of cycle-extendable graphs

Innovation

• Characterizes planar cycle-extendable graphs comprehensively
• Uses K2 and four infinite families for classification
• Builds on prior claw-free and bipartite graph studies