It is well-known that any class of simple graphs, that is characterized by finitely many forbidden minors, also admits a characterization by finitely many forbidden topological minors; furthermore, the list of forbidden topological minors may be derived from the list of forbidden minors. We prove a similar result in Matching Theory. Our Main Theorem states that any class of matching covered graphs, that is characterized by finitely many forbidden $S$-minors that are cubic, also admits a characterization by finitely many forbidden conformal minors that are cubic as well; once again, the list of forbidden conformal minors may be derived from the list of forbidden $S$-minors. In order to establish the above, we first prove that every matching covered graph has one of two graphs as a conformal minor -- either $K_4$, or the $Θ$ graph (that is, two vertices joined by three edges). (In fact, we need and prove a much stronger statement.) This is reminiscent of a theorem due to Lovász: every nonbipartite matching covered graph has one of two graphs as a conformal minor -- either $K_4$, or the triangular prism $\overline{C_6}$. As applications of our Main Theorem, we deduce known 'forbidden conformal minor characterizations' of pfaffian near-bipartite graphs, and of pfaffian solid graphs, using their respective known 'forbidden $S$-minor characterizations'.

翻译：众所周知，任何以有限个禁止子式为特征的单图类，同样允许以有限个禁止拓扑子式为特征；此外，禁止拓扑子式列表可由禁止子式列表推导得出。我们在匹配理论中证明了类似结果。我们的主要定理指出：任何以有限个三次禁止$S$-子式为特征的匹配覆盖图类，同样允许以有限个三次禁止共形子式为特征；同样地，禁止共形子式列表可由禁止$S$-子式列表推导得出。为建立上述结果，我们首先证明每个匹配覆盖图必以$K_4$或$\Theta$图（即由三条边连接的两个顶点）作为共形子式。（事实上，我们需要并证明了一个更强的结论。）这令人联想到Lovász定理：每个非二分匹配覆盖图必以$K_4$或三棱柱图$\overline{C_6}$作为共形子式。作为主要定理的应用，我们利用Pfaffian近二分图与Pfaffian固态图各自已知的"禁止$S$-子式特征"，推导出它们已知的"禁止共形子式特征"。