Bricks in which every vertex is incident with a forcing edge

Abstract

An edge of a matching covered graph G is a forcing edge if it lies in precisely one perfect matching of G. A matching covered graph is a brick if and only if it is 3-connected and bicritical (the deletion of each pair of distinct vertices results in a graph with a perfect matching). In this paper, we prove that every vertex of a brick is incident with a forcing edge if and only if the brick is an odd wheel up to multiple edges.