Equivalence classes in matching covered graphs

A connected graph $G$, of order two or more, is matching covered if each edge lies in some \pema. The tight cut decomposition of a matching covered graph $G$ yields a list of bricks and braces; as per a theorem of Lov{\'a}sz~\cite{lova87}, this list is unique (up to multiple edges); $b(G)$ denotes the number of bricks, and $c_4(G)$ denotes the number of braces that are isomorphic to the cycle $C_4$ (up to multiple edges). Two edges $e$ and $f$ are mutually dependent if, for each perfect matching $M$, $e \in M$ if and only if $f \in M$; Carvalho, Lucchesi and Murty investigated this notion in their landmark paper~\cite{clm99}. For any matching covered graph $G$, mutual dependence is an equivalence relation, and it partitions $E(G)$ into equivalence classes; this equivalence class partition is denoted by $\mathcal{E}_G$ and we refer to its parts as equivalence classes of $G$; we use $\varepsilon(G)$ to denote the cardinality of the largest equivalence class. The operation of `splicing' may be used to construct bigger matching covered graphs from smaller ones; see~\cite{lckm18}; `tight splicing' is a stronger version of `splicing'. (These are converses of the notions of `separating cut' and `tight cut'.) In this article, we answer the following basic question: if a matching covered graph $G$ is obtained by `splicing' (or by `tight splicing') two smaller matching covered graphs, say~$G_1$~and~$G_2$, then how is $\mathcal{E}_G$ related to $\mathcal{E}_{G_1}$ and to $\mathcal{E}_{G_2}$ (and vice versa)? As applications of our findings: firstly, we establish tight upper bounds on $\varepsilon(G)$ in terms of $b(G)$ and $c_4(G)$; secondly, we answer a recent question of He, Wei, Ye and Zhai~\cite{hwyz19}, in the affirmative, by constructing graphs that have arbitrarily high $\kappa(G)$~and~$\varepsilon(G)$ simultaneously, where $\kappa(G)$ denotes the vertex-connectivity.