Near-bipartite bricks in which every b-invariant edge is a forcing edge
Abstract
A connected graph is matching covered if it has at least one edge and every edge lies in some perfect matching.Lovász proved that every matching covered graph G can be uniquely decomposed into a list of bricks and braces up to multiple edges. Denote by b(G) the number of bricks in such a decomposition. An edge e of G is removable if G-e is also matching covered; is b-invariant if e is removable and b(G-e)=b(G). Furthermore, an edge e of G is a forcing edge if it lies in precisely one perfect matching of G. Lucchesi and Murty proposed the problem of characterizing bricks, distinct from K4, C6, and the Petersen graph, in which every b-invariant edge is a forcing edge. In this paper, we solve this problem for near-bipartite bricks by providing a complete characterization.
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