# $b$-invariant edges in essentially 4-edge-connected near-bipartite cubic bricks

## TL;DR

This paper proves a conjecture that essentially 4-edge-connected near-bipartite cubic bricks have at least half as many b-invariant edges as their number of vertices, expanding understanding of their structure.

## Contribution

It confirms the conjecture that such bricks have at least |V(G)|/2 b-invariant edges and characterizes those attaining this bound.

## Key findings

- Confirmed the conjecture for near-bipartite bricks
- Identified bricks with the minimum number of b-invariant edges
- Extended previous results to a broader class of graphs

## Abstract

A {\em brick} is a non-bipartite matching covered graph without non-trivial tight cuts. Bricks are building blocks of matching covered graphs. We say that an edge $e$ in a brick $G$ is {\em $b$-invariant} if $G-e$ is matching covered and a tight cut decomposition of $G-e$ contains exactly one brick. A 2-edge-connected cubic graph is {\em essentially 4-edge-connected} if it does not contain nontrivial 3-cuts. A brick $G$ is {\em near-bipartite} if it has a pair of edges $\{e_1, e_2\}$ such that $G-\{e_1,e_2\}$ is bipartite and matching covered. Kothari, de Carvalho, Lucchesi and Little proved that each essentially 4-edge-connected cubic non-near-bipartite brick $G$, distinct from the Petersen graph, has at least $|V(G)|$ $b$-invariant edges. Moreover, they made a conjecture: every essentially 4-edge-connected cubic near-bipartite brick $G$, distinct from $K_4$, has at least $|V(G)|/2$ $b$-invariant edges. We confirm the conjecture in this paper. Furthermore, all the essentially 4-edge-connected cubic near-bipartite bricks, the numbers of $b$-invariant edges of which attain the lower bound, are presented.