Minimal bricks with the maximum number of edges

Abstract

A 3‐connected graph is a brick if, after the removal of any two distinct vertices, the resulting graph has a perfect matching. A brick is minimal if, for every edge , deleting results in a graph that is not a brick. Norine and Thomas proved that every minimal brick with vertices, which is distinct from the prism or the wheel on four, six, or eight vertices, has at most edges. In this paper, we characterize the extremal minimal bricks with vertices that meet this upper bound, and we prove that the number of extremal graphs equals if , 5 if , 10 if and 0 if , respectively.