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.

翻译：若一个非平凡连通图的每条边都属于某个完美匹配，则称其为匹配覆盖图。对于大多数涉及完美匹配的问题，可将研究范围限制在匹配覆盖图上；因此，现有大量关于此类图的文献。该理论的基石是Lovász和Plummer提出的耳分解定理。他们的定理是解决问题的基本工具，同时也引出了若干有趣的开放问题；下文将讨论其中两个问题，并解决其中之一。若图$G$的子图$H$满足$G-V(H)$存在完美匹配，则称$H$是保形的。这一概念与前述耳分解定理本质相关——该定理意味着每个匹配覆盖图（除$K_2$和偶环外）都包含一个保形的θ双细分子图，或一个保形的$K_4$双细分子图，或两者兼有。（此处θ指由两个顶点通过三条边连接的图。）这直接引出了两个问题：刻画无θ（以及无$K_4$）的匹配覆盖图。Kothari和Murty[J. Graph Theory, 82 (1), 2016]给出了平面无$K_4$匹配覆盖图的刻画；非平面情形仍为开放问题。本文提出了无θ匹配覆盖图的完整刻画，该刻画可直接导出对应判定问题的多项式时间算法。我们的刻画主要依赖于Edmonds、Lovász和Pulleyblank[Combinatorica, 2, 1982]关于匹配覆盖图紧割分解理论的奠基性成果。作为推论，我们给出了无θ图规模的两个上界：$m\leq 2n-1$与$m\leq \frac{3n}{2}+b-1$，其中$b$表示该图任意紧割分解得到的砖块数量；针对每个上界，我们给出了达到紧界的极图刻画。Petersen图与$K_4$在我们的结果中起着关键作用。