Removable edges in near-bipartite bricks

TL;DR

This paper investigates removable edges in near-bipartite bricks, establishing lower bounds on the number of removable edges and characterizing graphs that attain this bound, advancing understanding of their structural properties.

Contribution

It provides a lower bound on removable edges in near-bipartite bricks and characterizes all graphs that achieve this bound, extending prior work on brick structures.

Findings

Every vertex, except at most six of degree three in two triangles, is incident with at most two nonremovable edges.

Near-bipartite bricks have at least (|V(G)| - 6)/2 removable edges.

Characterization of graphs attaining the lower bound on removable edges.

Abstract

An edge $e$ of a matching covered graph $G$ is removable if $G-e$ is also matching covered. The notion of removable edge arises in connection with ear decompositions of matching covered graphs introduced by Lov\'asz and Plummer. A nonbipartite matching covered graph $G$ is a brick if it is free of nontrivial tight cuts. Carvalho, Lucchesi, and Murty proved that every brick other than $K_4$ and $\overline{C_6}$ has at least $\Delta-2$ removable edges. A brick $G$ is near-bipartite if it has a pair of edges $\{e_1,e_2\}$ such that $G-\{e_1,e_2\}$ is a bipartite matching covered graph. In this paper, we show that in a near-bipartite brick $G$ with at least six vertices, every vertex of $G$, except at most six vertices of degree three contained in two disjoint triangles, is incident with at most two nonremovable edges; consequently, $G$ has at least $\frac{|V(G)|-6}{2}$ removable edges. Moreover, all graphs attaining this lower bound are characterized.