Wheel-like bricks and minimal matching covered graphs

TL;DR

This paper characterizes wheel-like bricks in matching covered graphs, showing they can be constructed from odd wheel graphs, and establishes degree bounds for minimal matching covered graphs.

Contribution

It provides a structural characterization of wheel-like bricks and determines degree bounds for minimal matching covered graphs beyond K2.

Findings

Wheel-like bricks can be obtained by splicing odd wheel graphs.

Minimum degree of minimal matching covered graphs (not K2) is 2 or 3.

Characterization advances understanding of matching covered graph structures.

Abstract

A connected graph G with at least two vertices is matching covered if each of its edges lies in a perfect matching. We say that an edge e in a matching covered graph G is removable if G-e is matching covered. A pair {e; f} of edges of a matching covered graph 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, introduced by Lovasz and Plummer in connection with ear decompositions of matching covered graphs. A 3-connected graph is a brick if the removal of any two distinct vertices, the left graph has a perfect matching. 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 proposed a problem of characterizing wheel-like bricks. We show that every wheel-like brick may be obtained by splicing graphs whose underlying simple graphs are odd wheels in a certain manner. A matching covered graph is minimal if the removal of any edge, the left graph is not matching covered. Lovasz and Plummer proved that the minimum degree of a minimal matching covered bipartite graph different from K2 is 2 by ear decompositions in 1977. By the properties of wheel-like bricks, we prove that the minimum degree of a minimal matching covered graph other than K2 is 2 or 3.