this paper we show an information-theoretic lower bound of kn o(kn) on the minimum number of bits to represent an unlabeled connected n-node graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn +2m+o(n) bits, that is 4kn +2n k

Let G be an n-node planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility representation of G. Given a canonical ordering of the triangulated G, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained �from Schnyder’s � realizer for the triangulated G yields a visibility representation of G no wider than 22n−40. Our easy-to-implement O(n)-time algorithm bypasses the complicated subroutines for 15 four-connected components and four-block trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant’s open question about whether � � 3n−6 is a 2 worst-case lower bound on the required width. Also, if G has no degree-three (respectively, degreefive) internal node, then our visibility representation for G is no wider than � �

Floor-planning is a fundamental step in VLSI chip design. Based upon the concept of orderly spanning trees, we present a simple O(n)-time algorithm to construct a oor-plan for any n-node plane triangulation. In comparison with previous oor-planning algorithms in the iterature, our solution is not only simpler in the algorithm itself, but also produces oor-plans which require fewer module types. An equally important aspect of our new algorithm lies in its ability to fit the floor-plan area in a rectangle of size (n-1) by (2n+1)/3.

We propose a new linear time algorithm to represent a planar graph. Based on a speci c triangulation of the graph, our coding takes on average 5:03 bits per node, and 3:37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n nodes is at most 2 n+O(log n) , where 5:007.

In this paper we study connections between planar graphs, Schnyder woods, and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and the dimension theory of orders. Orthogonal surfaces explain connections between these seemingly unrelated notions. We use these connections for an intuitive proof of the Brightwell-Trotter Theorem which says, that the face lattice of a 3-polytope minus one face has order dimension three. Our proof yields a linear time algorithm for the construction of the three linear orders that realize the face lattice. Coplanar orthogonal surfaces are in correspondence with a large class of convex straight line drawings of 3-connected planar graphs. We show that Schnyder’s face counting approach with weighted faces can be used to construct all coplanar orthogonal surfaces and hence the corresponding drawings. Appropriate weights are computable in linear time. 1

Schnyder woods are a well known combinatorial structure for planar graphs, which yields a decomposition into 3 vertexspanning trees. Our goal is to extend definitions and algorithms for Schnyder woods designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere, i.e., of genus 0) to the more general case of graphs embedded on surfaces of arbitrary genus. First, we define a new traversal order of the vertices of a triangulated surface of genus g together with an orientation and coloration of the edges that extends the one proposed by Schnyder for the planar case. As a by-product we show how some recent schemes for compression and compact encoding of graphs can be extended to higher genus. All the algorithms presented here have linear time complexity.

A plane pseudograph is a plane graph allowing both loops and multiple edges. The encoding technique we propose belongs to a class of compression methods based on a deterministic graph traversal, which is encoded as a bit string. Arrays of vertices and edges stored in the order defined by this traversal, together with the bit string, allow the retrieval of the original graph. Within this framework, the most general methods to encode plane graphs allow obtaining the cyclic ordering of the edges incident to each vertex. Provided any directed edge incident to the infinite face is specified, the plane graph is uniquely defined. However, this information may not be sufficient to re-create the faces of the original graph when loops are allowed. In this paper, we analyse what information must be encoded in the bit string to retrieve correctly all incidences among the vertices, edges and faces of any plane pseudograph. Let G be a connected plane pseudograph with V vertices, E edges and F faces. The most common representation of G in computational geometry is the half-edge data structure, which requires 2E log V +(V +4E +F) log(2E)+2E log F bits to store the connectivity of the graph. The compression method proposed in this paper encodes the graph connectivity in 4E + 1 bits, and allows encoding-decoding the data structure representing the graph in O(E) time.

Canonical ordering is an important tool in planar graph drawing and other applications. Although a linear-time algorithm to determine canonical orderings has been known for a while, it is rather complicated to understand and implement, and the output is not uniquely determined. We present a new approach that is simpler and more intuitive, and that computes a newly defined leftist canonical ordering of a triconnected graph which is a uniquely determined leftmost canonical ordering. Further, we discuss duality aspects and relations to Schnyder woods. Submitted:

Abstract. Canonical ordering is an important tool in planar graph drawing and other applications. Although a linear-time algorithm to determine canonical orderings has been known for a while, it is rather complicated to understand and implement, and the output is not uniquely determined. We present a new approach that is simpler and more intuitive, and that computes a newly defined leftist canonical ordering of a triconnected graph which is a uniquely determined leftmost canonical ordering. 1