  Let $G$ be a matching-covered graph, i.e., every edge is contained in a
perfect matching. An edge subset $X$ of $G$ is feasible if there exists two
perfect matchings $M_1$ and $M_2$ such that $|M_1\cap X|\not\equiv |M_2\cap X|
\pmod 2$. Lukot'ka and Rollov\'a proved that an edge subset $X$ of a regular
bipartite graph is not feasible if and only if $X$ is switching-equivalent to
$\emptyset$, and they further ask whether a non-feasible set of a regular graph
of class 1 is always switching-equivalent to either $\emptyset$ or $E(G)$? Two
edges of $G$ are equivalent to each other if a perfect matching $M$ of $G$
either contains both of them or contains none of them. An equivalent class of
$G$ is an edge subset $K$ with at least two edges such that the edges of $K$
are mutually equivalent. An equivalent class is not a feasible set. Lov\'asz
proved that an equivalent class of a brick has size 2. In this paper, we show
that, for every integer $k\ge 3$, there exist infinitely many $k$-regular
graphs of class 1 with an arbitrarily large equivalent class $K$ such that $K$
is not switching-equivalent to either $\emptyset$ or $E(G)$, which provides a
negative answer to the problem proposed by Lukot'ka and Rollov\'a. Further, we
characterize bipartite graphs with equivalent class, and characterize
matching-covered bipartite graphs of which every edge is removable.
