Abstract
AbstractA 3‐connected graph is a brick if, after the removal of any two distinct vertices, the resulting graph has a perfect matching. Lovász proved that the dimension of the matching lattice of a brick is equal to . We say a brick is extremal if the number of perfect matchings in is exactly . de Carvalho et al. characterized extremal bricks and conjectured that every extremal nonsolid brick other than the Petersen graph is the result of the splicing of an extremal brick and a , up to multiple edges. In this paper, we present an infinite family of graphs showing that this conjecture fails.
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