On Planar Extremal Nonsolid Bricks

ABSTRACT

A 3‐connected graph is called a brick when the deletion of any two distinct vertices leaves a graph with a perfect matching. Lovász (J. Combin. Theory (B) 43 (1987), 187–222.) proved that each brick contains at least perfect matchings. We call a brick extremal when its number of perfect matchings attains this lower bound. De Carvalho, Lucchesi, and Murty (J. Graph Theory 48 (2005), 19–50.) conjectured that every nonsolid extremal brick other than the Petersen graph is obtained, up to multiple edges, by splicing of an extremal brick and a . Feng and Lu (J. Graph Theory 97 (2021), 189–193.) constructed an infinite non‐planar family showing that the conjecture is false. In this paper, we prove that the conjecture does hold for planar extremal nonsolid bricks.