Equivalence classes in matching covered graphs

Let $G$ be a matching covered graph. Two edges $ e_1$ and $e_2$ of $G$ are {\it equivalent} if $\{e_1,e_2\}\subseteq M$ or $\{e_1,e_2\}\cap M=\emptyset$ for every perfect matching $M$ of $G$. An {\it equivalent class} of $G$ is an edge set $S$ of $G$ with at least two edges such that the edges of $S$ are mutually equivalent. Lovasz gave a characterization of the equivalence classes in a brick: if $S$ is an equivalence class of a brick $G$, then $|S|= 2$ and $G-S$ is bipartite. For matching covered graphs, there are infinitely many 2-connected graphs with an arbitrarily large equivalent class. Recently, He et al. [Journal of Graph Theory, DOI: 10.1002/jgt.22411, 2018] asked whether there exist infinitely many 3-connected matching covered graphs with an arbitrarily large equivalent class. By constructing infinitely many $k$ $(k\geq 3)$-connected matching covered graphs with an arbitrarily large equivalent class, we give a positive answer to this problem. Moreover, we consider the equivalent classes in cubic graphs. If $G$ is a 3-connected cubic matching covered graph, $E_0$ is an equivalent class of $G$, then $|E_0|\leq 2b(G)$, where $b(G)$ is the brick number of $G$.