Planar Cycle-Extendable Graphs

• Main Authors: Dalwadi, Aditya Y, Pause, Kapil R Shenvi, Diwan, Ajit A, Kothari, Nishad
• Format: Journal Article
• Language: English
• Published: 24-05-2024
• Subjects: Computer Science - Discrete Mathematics; Mathematics - Combinatorics
• DOI: 10.48550/arxiv.2405.15416

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.