Leapfrog transformations and polyhedra of Clar type

Abstract

The so-called leapfrog transformation that was first introduced for fullerenes (trivalent polyhedra with 12 pentagonal faces and all other faces hexagonal) is generalised to general polyhedra and maps on surfaces. All spherical polyhedra can be classified according to their leapfrog order. A polyhedron is said to be of Clar type if there exists a set of faces that cover each vertex exactly once. It is shown that a fullerence is of Clar type if and only if it is a leapfrog transform of another fullerene. Several basic transformations on maps are defined by means of which the leapfrog and other transformations can be accomplished.