Planar cycle-extendable graphs

TL;DR

This paper characterizes all planar cycle-extendable graphs, which are graphs where even cycles extend to perfect matchings, using $K_2$ and four infinite families, advancing understanding of their structure.

Contribution

It provides a complete characterization of planar cycle-extendable graphs in terms of basic building blocks and infinite families, filling a gap in the existing literature.

Findings

Characterization of all planar cycle-extendable graphs.

Identification of $K_2$ and four infinite families as fundamental components.

Extension of previous results on claw-free and bipartite planar graphs.

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 Lov\'asz 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, 2022, provided a characterization of claw-free cycle-extendable graphs. Guo and Zhang, 2004, and independently Zhang and Li, 2012, 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.