# Equivalence classes in matching covered graphs

## TL;DR

This paper investigates the structure of equivalence classes of edges in matching covered graphs, especially how they behave under splicing operations, and establishes bounds and constructions related to these classes and graph connectivity.

## Contribution

It characterizes the relationship of equivalence classes in matching covered graphs under splicing, providing bounds and answering open questions on graph connectivity and class sizes.

## Key findings

- Established bounds on the size of the largest equivalence class
- Analyzed how equivalence classes change under splicing operations
- Constructed graphs with high vertex-connectivity and large equivalence classes

## Abstract

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.