Planar Cycle-Extendable Graphs
<|reference_start|>Planar Cycle-Extendable Graphs: 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.<|reference_end|>
@article{dalwadi2024planar,
    title={Planar Cycle-Extendable Graphs},
    author={Aditya Y Dalwadi, Kapil R Shenvi Pause, Ajit A Diwan, Nishad Kothari},
    journal={arXiv preprint arXiv:2405.15416},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.15416},
    primaryClass={math.CO cs.DM}
}