We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for many graph drawing problems. -- Every triconnected planar graph G can be drawn convexly with straight lines on an (2n \Gamma 4) \Theta (n \Gamma 2) grid, where n is the number of vertices. -- Every triconnected planar graph with maximum degree four can be drawn orthogonally on an n \Theta n grid with at most d 3n 2 e + 4, and if n ? 6 then every edge has at most two bends. -- Every 3-planar graph G can be drawn with at most b n 2 c + 1 bends on an b n 2 c \Theta b n 2 c grid. -- Every triconnected planar graph G can be drawn planar on an (2n \Gamma 6) \Theta (3n \Gamma 9) grid with minimum angle larger than 2 d radians and at most 5n \Gamma 15 bends, with d the maximum d...

In this paper, NP-complete and polynomial solvable problems are reduced to the SATISFIABILITY problem. We call this process "programming" in propositional logic. On the one hand, the programs (propositional formulas) derived by this process build a rich pool of easy and hard (non-random) formulas for SATISFIABILITY-solving heuristics. On the other hand, the implementations (programs) give rise to new heuristics for solving SATISFIABILITY. Contents 1 Introduction 1 2 Useful formulas 2 3 Useful techniques 3 3.1 Removing terms of the form ": : : =) V ::: : : :" : : : : : : : : : : : : : : : : : : : : : 3 3.2 Removing terms of the form ": : : W ::: V ::: (: : :)" : : : : : : : : : : : : : : : : : : : : 3 3.3 Removing terms of the form ": : : () : : :" : : : : : : : : : : : : : : : : : : : : : : : 3 4 Different Reductions 4 4.1 Problems from P : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 4.1.1 SHORTEST PATH / SAT : : : : : : : : : : : : : : : : : :...

A graph is planar if it can be drawn on the plane with vertices at unique locations and no edge intersections except at the vertex endpoints. Due to the wealth of interest from the computer science community, there are a number of remarkable but complex O(n) planar embedding algorithms. This paper presents an O(n) planar embedding algorithm that avoids a number of the complexities of prior approaches (an early version of this work was presented at the January 1999 Symposium on Discrete Algorithms).