Planar wheel-like bricks

TL;DR

This paper proves that every planar wheel-like brick is an odd wheel, confirming a conjecture related to the structure of certain nonbipartite matching covered graphs.

Contribution

It provides a proof of Lucchesi and Murty's conjecture that all planar wheel-like bricks are odd wheels.

Findings

Confirmed that all planar wheel-like bricks are odd wheels

Established structural properties of wheel-like bricks in matching covered graphs

Contributed to the theory of ear decompositions in graph theory

Abstract

An edge e in a matching covered graph G is removable if G-e is matching covered; a pair {e; f} of edges of G is a removable doubleton if G-e-f is matching covered, but neither G-e nor G-f is. Removable edges and removable doubletons are called removable classes, which was introduced by Lovasz and Plummer in connection with ear decompositions of matching covered graphs. A brick is a nonbipartite matching covered graph without nontrivial tight cuts. A brick G is wheel-like if G has a vertex h, such that every removable class of G has an edge incident with h. Lucchesi and Murty conjectured that every planar wheel-like brick is an odd wheel. We present a proof of this conjecture in this paper.