Canonical Decomposition, Realizer, Schnyder Labeling and Orderly Spanning Trees of Plane Graphs (Extended Abstract) Kazuyuki Miura, Machiko Azuma, and Takao Nishizeki Graduate School of Information Sciences Tohoku University, Sendai 980-8579, Japan {miura,azuma}@nishizeki.ecei.tohoku.ac.jp nishi@ecei.tohoku.ac.jp 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. 1 Introduction Recently automatic aesthetic drawing of graphs has created intense interest due to their broad applications, and as a consequence, a number of drawing methods have come out [3–9, 11–13, 15, 18, 21, 23]. The most typical drawing of a plane graph G is the straight line drawing in which all vertices of G are drawn as points and all edges are drawn as straight line segments without any edge-intersection. A straight line drawing of G is called a grid drawing if the vertices of G are put on grid points of integer coordinates. A straight line drawing of G is called a convex drawing if every face boundary is drawn as a convex polygon [4, 22,23]. A convex drawing of G is called an outer triangular convex drawing if the outer face boundary is drawn as a triangle, as illustrated in Fig. 1. A canonical decomposition, a realizer, a Schnyder labeling and an orderly spanning tree of a plane graph G play an important role in straight-line draw- ings, convex grid drawings, floor-plannings, graph encoding, etc. [1, 2, 5–8,12, 13, 15–18, 21]. It is known that the triconnectivity is a sufficient condition for their existence in G [5, 10, 12, 21], 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 K.-Y. Chwa and J.I. Munro (Eds.): COCOON 2004, LNCS 3106, pp. 309–318, 2004. c  Springer-Verlag Berlin Heidelberg 2004