vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1

In this paper we introduce a new drawing style of a plane graph G, called proper box rectangular (PBR ) drawing. It is defined to be a drawing of G such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal or a vertical line segment, and each face is drawn as a rectangle. We establish necessary and sufficient conditions for G to have a PBR drawing. We also give a simple linear time algorithm for finding such drawings. The PBR drawing is closely related to the box rectangular (BR ) drawing defined by Rahman, Nakano and Nishizeki [17]. Our method can be adapted to provide a new simpler algorithm for solving the BR drawing problem. 1 Introduction The problem of "nicely" drawing a graph G has received increasing attention [5]. Typically, we want to draw the edges and the vertices of G on the plane so that certain aesthetic quality conditions and/or optimization measures are met. Such drawings are very useful in visualizing planar graphs and fi...

An orthogonal drawing of a graph is an embedding of the graph in the two-dimensional grid such that edges are routed along grid-lines. In this paper we explore lower bounds for orthogonal graph drawings. We prove lower bounds on the number of bends and, when crossings are not allowed, also lower bounds on the size of the grid. Communicated by D. Wagner: submitted July 1997; revised November 1998. Some results of this paper were part of the author's diploma thesis at TU Berlin under the supervision of Prof. R. Mohring, and have been presented in an extended abstract at Graph Drawing '95, Passau, Germany. T. Biedl., New Lower Bounds, JGAA, 2(7) 1-31 (1998) 2 1 Introduction A graph G = (V; E) is an abstract structure consisting of points (or vertices) V and connections (or edges) E. Such a structure is found in many industrial applications, such as networks, production schedules and diagrams. With the aid of graph drawing, a graph is displayed in visual form, and the underlying infor...

x 1 Introduction 1 1.1 Historical Background . . .............................. 4 1.2 Drawing Styles . ................................... 4 1.2.1 Polyline drawings .............................. 5 1.2.2 Planar drawings ............................... 5 1.2.3 Straight line drawings ............................ 6 1.2.4 Orthogonal drawings . . ........................... 7 1.2.5 Grid drawings ................................ 8 1.3 Properties of Drawings ................................ 9 1.4 Scope of this Thesis .................................. 10 1.4.1 Rectangular drawings . . . ......................... 11 1.4.2 Orthogonal drawings . . ........................... 12 1.4.3 Box-rectangular drawings ........................... 14 1.4.4 Convex drawings . . ............................. 16 1.5 Summary ....................................... 16 2 Preliminaries 20 2.1 Basic Terminology .................................. 20 2.1.1 Graphs and Multigraphs ........................... 20 i CO...

An orthogonal drawing of a plane graph G is a drawing of G in which each edge is drawn as a sequence of alternate horizontal and vertical line segments. In this paper we give a linear-time algorithm to find an orthogonal drawing of a given 3-connected cubic plane graph with the minimum number of bends. The best previously known algorithm takes time O(n 7/4 # log n) for any plane graph with n vertices. Communicated by Giuseppe Di Battista and Petra Mutzel.

Abstract. Schnyder woods are decompositions of simple triangulations into three edge-disjoint spanning trees crossing each other in a specific way. In this article, we define a generalization of Schnyder woods to d-angulations (plane graphs with faces of degree d) for all d ≥ 3. A Schnyder decomposition is a set of d spanning forests crossing each other in a specific way, and such that each internal edge is part of exactly d − 2 of the spanning forests. We show that a Schnyder decomposition exists if and only if the girth of the d-angulation is d. As in the case of Schnyder woods (d = 3), there are alternative formulationsintermsoforientations (“fractional ” orientations when d ≥ 5)and in terms of corner-labellings. Moreover, the set of Schnyder decompositions on a fixed d-angulation of girth d is a distributive lattice. We also show that the structures dual to Schnyder decompositions (on d-regular plane graphs of mincut d rooted at a vertex v ∗ ) are decompositions into d spanning trees rooted atv ∗ such that each edge not incidentto v ∗ isused in opposite directions by two trees. Additionally, for even values of d, we show that a subclass of Schnyder decompositions, which are called even, enjoy additional properties that yield a reduced formulation; in the case d = 4, these correspond to wellstudied structures on simple quadrangulations (2-orientations and partitions into 2 spanning trees). In the case d = 4, the dual of even Schnyder decompositions yields (planar) orthogonal and straight-line drawing algorithms. For a 4-regular plane graph G of mincut 4 with n vertices plus a marked vertex v, the vertices of G\v are placed on a (n−1)×(n−1) grid according to a permutation pattern, and in the orthogonal drawing each of the 2n−2 edges of G\v has exactly one bend. Embedding also the marked vertex v is doable at the cost of two additional rows and columns and 8 additional bends for the 4 edges incident to v. We propose a further compaction step for the drawing algorithm and show that the obtained grid-size is strongly concentrated around 25n/32 ×25n/32 for a uniformly random instance with n vertices. 1.