  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.
