Diagonal flips in plane graphs with triangular and quadrangular faces

A tri-quadrangulation is a connected simple plane graph with each face either triangular or quadrangular. Recently, Aichholzer et al. (2014) proved that any two tri-quadrangulations with n vertices and m≥2 triangular faces can be transformed into each other by a sequence of local transformations, called a diagonal flip, and their algorithm guarantees that at most O(n2) diagonal flips are sufficient. In this paper, we improve their upper bound to O(n), and prove that the linear order of the estimation is best possible.