On graphs with $1$-matching and $2$-matching edges

Let $G$ be a graph admitting a perfect matching. An edge is called a {\it $k$-matching edge} if it belongs to exactly $k$ perfect matchings, and a {\it $k^{+}$-matching edge} if it belongs to at least $k$ perfect matchings. Thus, {\it an admissible edge} is a $1^{+}$-matching edge, and a connected graph is {\it matching covered} if every edge is admissible. We call a connected graph {\it $k$-matching covered} if every edge is a $k$-matching edge; in particular, a $2$-matching covered graph is called {\it matching double covered}. Motivated by matching-covered graph theory and the Berge--Fulkerson conjecture (1970s), we introduce the class $\mathfrak{B}$ of connected graphs in which every edge is either a $1$-matching edge or a $2$-matching edge, and no perfect matching contains edges of both types. In particular, every matching double covered graph belongs to $\mathfrak{B}$. Using ear decompositions and tight-cut decompositions, we establish a complete structural characterization of graphs in $\mathfrak{B}$. These characterizations reveal how restrictions on the number of perfect matchings containing each edge determine the global structure of the corresponding matching-covered graphs.