Cubic bricks that every b-invariant edge is forcing

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.