Let $G=(V,E)$ be a matching-covered graph and $X$ be an edge set of $G$. $X$ is said to be feasible if there exist two perfect matchings $M_1$ and $M_2$ in $G$ such that $|M_1\cap X|\not \equiv|M_2\cap X|\ (\mbox{mod } 2)$. For any $V_0\subseteq V$, $X$ is said to be switching-equivalent to $X\oplus \nabla_G(V_0)$, where $\nabla_G(V_0)$ is the set of edges in $G$ each of which has exactly one end in $V_0$ and $A \oplus B$ is the symmetric difference of two sets $A$ and $B$. Lukot'ka and Rollov\'a showed that when $G$ is regular and bipartite, $X$ is non-feasible if and only if $X$ is switching-equivalent to $\emptyset$. This article extends Lukot'ka and Rollov\'a's result by showing that this conclusion holds as long as $G$ is matching-covered and bipartite. This article also studies matching-covered graphs $G$ whose non-feasible edge sets are switching-equivalent to $\emptyset$ or $E$ and partially characterizes these matching-covered graphs in terms of their ear decompositions. Another aim of this article is to construct infinite many $r$-connected and $r$-regular graphs of class 1 containing non-feasible edge sets not switching-equivalent to either $\emptyset$ or $E$ for an arbitrary integer $r$ with $r\ge 3$, which provides negative answers to problems asked by Lukot'ka and Rollov\'a and He, et al respectively.

中文翻译：

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关于匹配覆盖图中的不可行边集

令 $G=(V,E)$ 为匹配覆盖图，$X$ 为 $G$ 的边集。如果 $G$ 中存在两个完美匹配 $M_1$ 和 $M_2$ 使得 $|M_1\cap X|\not \equiv|M_2\cap X|\ (\mbox{mod 2)$。对于任何 $V_0\subseteq V$，$X$ 被说成是切换等价于 $X\oplus \nabla_G(V_0)$，其中 $\nabla_G(V_0)$ 是 $G$ 中每条边的集合$V_0$ 和 $A \oplus B$ 正好有一个端点是两个集合 $A$ 和 $B$ 的对称差。Lukot'ka 和 Rollov\'a 表明，当 $G$ 是正则和二分时，$X$ 是不可行的，当且仅当 $X$ 是切换等价于 $\emptyset$。本文扩展了 Lukot'ka 和 Rollov\'a 的结果，表明只要 $G$ 是匹配覆盖且二分的，这个结论就成立。本文还研究了匹配覆盖图 $G$，其非可行边集切换等价于 $\emptyset$ 或 $E$，并根据耳分解部分描述了这些匹配覆盖图。这篇文章的另一个目的是构造无限多个 $r$-connected 和 $r$-regular graphs of class 1 包含不可行边集的不切换-等价于 $\emptyset$ 或 $E$ 的任意整数 $ r$ 和 $r\ge 3$，分别为 Lukot'ka 和 Rollov\'a 以及 He 等人提出的问题提供否定答案。