Let $G$ be a matching covered graph and let $X\subseteq E\left(G\right)$. Then $X$ is called feasible if there exist two perfect matchings $M_1$ and $M_2$ in $G$ such that $\mid M_1\cap X\mid \not\equiv \mid M_2\cap X\mid \left(\text{mod}2\right)$; otherwise, it is nonfeasible. For any vertex $v\in V\left(G\right)$, the switching operation of $X$ in $v$ is defined as the symmetric difference of $X$ and $\partial \left(v\right)$, where $\partial \left(v\right)$ denotes the set of all edges in $G$ incident with $v$. Two sets $X_1,X_2\subseteq E\left(G\right)$ are switching‐equivalent if $X_1$ can be obtained from $X_2$ by a series of switching operations and vice visa. Lukot'ka and Rollová showed that if $G$ is a regular bipartite graph, then $X$ is nonfeasible if and only if $X$ is switching‐equivalent to $\varnothing$ (as well as $E\left(G\right)$). In this paper, we give a complete characterization of nonfeasible sets in matching covered graphs. We prove that if $G$ is a matching covered graph and $X$ is an arbitrary nonfeasible set of $G$, then $X$ is switching‐equivalent to both $\varnothing$ and $E\left(G\right)$ if and only if $G$ is bipartite, and $X$ is switching‐equivalent to either $\varnothing$ or $E\left(G\right)$ (but not both) if and only if $G$ is nonbipartite and $G$ has an ear decomposition with exactly one double ear. We also show that for any two integers $s$ and $k$ with $s\ge 2$ and $k\ge \max \left\{3,s\right\}$, there exist infinitely many $k$‐regular simple graphs $G$ of class 1 with arbitrarily large brick number, connectivity $s$ and a nonfeasible set $X$ such that $X$ is not switching‐equivalent to either $\varnothing$ or $E\left(G\right)$. This gives negative answers to two problems proposed by Lukot'ka and Rollová and by He et al, respectively.

中文翻译：

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匹配覆盖图中不可行集的特征

让 $G$ 成为匹配的覆盖图，让 $X\subseteq Ë（G）$。然后$X$ 如果存在两个完美匹配，则称为可行 ${\mathrm{中号}}_{\mathrm{1个}}$ 和 ${\mathrm{中号}}_2$ 在 $G$ 这样 $\mid {\mathrm{中号}}_{\mathrm{1个}}\cap X\mid \not\equiv \mid {\mathrm{中号}}_2\cap X\mid （\text{模}2）$; 否则，这是不可行的。对于任何顶点$v\in V（G）$，切换操作 $X$ 在 $v$ 定义为的对称差 $X$ 和 $\partial （v）$，在哪里 $\partial （v）$ 表示的所有边的集合 $G$ 与 $v$。两套$X_{\mathrm{1个}}，X_2\subseteq Ë（G）$ 如果等于则切换 $X_{\mathrm{1个}}$ 可以从 $X_2$通过一系列转换操作和副签证。Lukot'ka和Rollová表明，如果$G$ 是规则的二部图，然后 $X$ 仅当且仅当不可行 $X$ 等效于 $\varnothing$ （以及 $Ë（G）$）。在本文中，我们对匹配的覆盖图中的不可行集进行了完整的刻画。我们证明$G$ 是匹配的覆盖图，并且 $X$ 是任意的不可行的 $G$， 然后 $X$ 两者都等效 $\varnothing$ 和 $Ë（G）$ 当且仅当 $G$ 是二分的 $X$ 切换到任一 $\varnothing$ 要么 $Ë（G）$ （但不是全部）当且仅当 $G$ 是非二分的 $G$耳朵分解成一只双耳。我们还表明对于任何两个整数$s$ 和 $ķ$ 与 $s\ge 2$ 和 $ķ\ge \mathrm{最高}\left\{3，s\right\}$，无限地存在 $ķ$常规简单图 $G$ 具有任意大砖块数，连通性的1类 $s$ 和一个不可行的集合 $X$ 这样 $X$ 不切换到任何一个 $\varnothing$ 要么 $Ë（G）$。这分别给卢科塔（Lukot'ka）和罗洛瓦（Rollová）以及He等人提出的两个问题给出了否定答案。