$\theta$-free matching covered graphs
<|reference_start|>$\theta$-free matching covered graphs: 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 $\theta$, or a conformal bisubdivision of $K_4$, possibly both. (Here, $\theta$ refers to the graph with two vertices joined by three edges.) This immediately leads to two problems: characterize $\theta$-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 $\theta$-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 $\theta$-free graph, namely, $m\leq 2n-1$ and $m\leq \frac{3n}{2}+b-1$, 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.<|reference_end|>
@article{joshi2024$\theta$-free,
    title={$\theta$-free matching covered graphs},
    author={Rohinee Joshi and Nishad Kothari},
    journal={arXiv preprint arXiv:2407.05264},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.05264},
    primaryClass={math.CO cs.DM}
}