Cubic bricks that every b-invariant edge is forcing

TL;DR

This paper characterizes all cubic bricks where every b-invariant edge is forcing, identifying exactly ten such graphs including well-known examples like K4, the prism, and the Petersen graph.

Contribution

It provides a complete classification of cubic bricks with the property that all b-invariant edges are forcing, solving a problem posed by Lucchesi and Murty.

Findings

Exactly ten cubic bricks have all b-invariant edges forcing.

Includes classical graphs like K4, the prism, and Petersen graph.

Provides a characterization distinct from other known brick classes.

Abstract

A connected graph G is matching covered if every edge lies in some perfect matching of G. Lovasz proved that every matching covered graph G can be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite) up to multiple edges. Denote by b(G) the number of bricks of G. An edge e of G is removable if G-e is also matching covered, and solitary (or forcing) if after the removal of the two end vertices of e, the left graph has a unique perfect matching. Furthermore, a removable edge e of a brick G is b-invariant if b(G-e) = 1. Lucchesi and Murty proposed a problem of characterizing bricks, distinct from K4, the prism and the Petersen graph, in which every b-invariant edge is forcing. We answer the problem for cubic bricks by showing that there are exactly ten cubic bricks, including K4, the prism and the Petersen graph, every b-invariant edge of which is forcing.