A note on removable edges in near-bricks

TL;DR

This paper extends the understanding of removable edges in matching covered graphs by generalizing a known result from bricks to a broader class called irreducible near-bricks, focusing on their structural properties.

Contribution

It generalizes the minimum number of removable edges from bricks to irreducible near-bricks, broadening the class of graphs for which this property is known.

Findings

Every irreducible near-brick different from K_4 and C_6 has at least Δ-2 removable edges.

The result applies to a wider class of graphs beyond traditional bricks.

Provides structural insights into near-bricks related to removable edges.

Abstract

An edge $e$ of a matching covered graph $G$ is removable if $G-e$ is also matching covered. Carvalho, Lucchesi, and Murty showed that every brick $G$ different from $K_4$ and $\overline{C_6}$ has at least $\Delta-2$ removable edges, where $\Delta$ is the maximum degree of $G$. In this paper, we generalize the result to irreducible near-bricks, where a graph is irreducible if it contains no single ear of length three or more.