Diagonal transformations in pentangulations on the sphere

An N -angulation is a finite simple plane graph such that each face is bounded by a cycle of length N , where N ≥ 3 is an integer. We consider diagonal transformations in N -angulations which consist of ⌊ 2 ⌋ kinds of operations. In the literature, Wagner proved that any two 3angulations (which are often called triangulations) with the same number of vertices can be transformed into each other by diagonal transformations. (In this case, a diagonal transformation in 3-angulations is unique, which is called a diagonal flip.) For 4-angulations (which are often called quadrangulations), Nakamoto proved the similar result. Moreover, the similar theorems were recently proved for N = 5 and 6. In the paper which includes the result for 6-angulations, the present author conjectured that any two N -angulations with the same number of vertices can be transformed into each other by diagonal transformations for any N ≥ 7. However, it was thought that the proof would be a routine with a case-by-case argument. Then, in this paper, we prove the conjecture by developing a more general technique for proving.