Abstract
This article is concerned with the counting and random sampling of plane graphs (simple planar graphs embedded in the plane). The main result of this study is a bijection between the class of plane graphs with triangular outer face, and a class of oriented binary trees. The number of edges and vertices of the plane graph can be tracked through the bijection. Consequently, the authors have obtained counting formulas and an efficient random sampling algorithm for rooted plane graphs (with arbitrary outer face) according to the number of edges and vertices. They also obtained a bijective link, via a bijection of Bona, between rooted plane graphs and 1342-avoiding permutations.