CANONICAL DECOMPOSITION, REALIZER, SCHNYDER LABELING AND ORDERLY SPANNING TREES OF PLANE GRAPHS

Abstract

A canonical decomposition, a realizer, a Schnyder labeling and an orderly spanning tree of a plane graph play an important role in straight-line grid drawings, convex grid drawings, floor-plannings, graph encoding, etc. It is known that the triconnectivity is a sufficient condition for their existence, but no necessary and sufficient condition has been known. In this paper, we present a necessary and sufficient condition for their existence, and show that a canonical decomposition, a realizer, a Schnyder labeling, an orderly spanning tree, and an outer triangular convex grid drawing are notions equivalent with each other. We also show that they can be found in linear time whenever a plane graph satisfies the condition.