An orthogonal drawing of a graph is an embedding in the plane such that all edges are drawn as sequences of horizontal and vertical segments. We present a linear time and space algorithm to draw any connected graph orthogonally on a grid of size n \Theta n with at most 2n + 2 bends. Each edge is bent at most twice. In particular for non-planar and non-biconnected planar graphs, this is a big improvement. The algorithm is very simple, easy to implement, and it handles both planar and non-planar graphs at the same time.

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...