A graph
G is matching‐covered if every edge of
G is contained in a perfect matching. A matching‐covered graph
G is strongly coverable if, for any edge
e of
G, the subgraph
G
\
e is still matching‐covered. An edge subset
X of a matching‐covered graph
G is feasible if there exist two perfect matchings
M
1 and
M
2 such that
∣
M
1
∩
X
∣
≢
∣
M
2
∩
X
∣0.3em
(
mod0.3em
2
), and an edge subset
K with at least two edges is an equivalent set if a perfect matching of
G contains either all edges in
K or none of them. A strongly matchable graph
G does not have an equivalent set, and any two independent edges of
G form a feasible set. In this paper, we show that for every integer
k
≥
3, there exist infinitely many
k‐regular graphs of class 1 with an arbitrarily large equivalent set that is not switching‐equivalent to either
∅ or
E
(
G
), which provides a negative answer to a problem of Lukot’ka and Rollová. For a matching‐covered bipartite graph
G
(
A
,
B
), we show that
G
(
A
,
B
) has an equivalent set if and only if it has a 2‐edge‐cut that separates
G
(
A
,
B
) into two balanced subgraphs, and
G
(
A
,
B
) is strongly coverable if and only if every edge‐cut separating
G
(
A
,
B
) into two balanced subgraphs
G
1
(
A
1
,
B
1
) and
G
2
(
A
2
,
B
2
) satisfies
∣
E
[
A
1
,
B
2
]
∣
≥
2 and
∣
E
[
B
1
,
A
2
]
∣
≥
2.