Minimal bricks with the maximum number of edges

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 e, deleting e results in a graph that is not a brick. Norine and Thomas proved that every minimal brick with 2n vertices, which is distinct from the prism or the wheel on four, six, or eight vertices, has at most 5n - 7 edges. In this paper, we characterize the extremal minimal bricks with 2n vertices that meet this upper bound, and we prove that the number of extremal graphs equals.(n - 1)(2)/4. if n >= 6, 5 if n = 5, 10 if n = 4 and 0 if n = 1, 2, 3, respectively.