Wagner's theorem on realizers

Abstract

Arealizer of a maximal plane graph is a set of three particular spanning trees. It has been used in several graph algorithms and particularly in graph drawing algorithms. We propose colored flips on realizers to generalize Wagner's theorem on maximal planar graphs to realizers. From this result, it is proved that ξ0 + ξ1 + ξ2 - Δ = n - 1 where ξi is the number of inner nodes in the tree Ti, Δ is the number of three colored faces in the realizer and n is the number of vertices. As an application of this formula, we show that orderly spanning trees with at most ⌊ 2n+1-Δ/3⌋ leaves can be computed in linear time. © 2002 Springer-Verlag Berlin Heidelberg.