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Drawing Planar Graphs Using the Canonical Ordering
 ALGORITHMICA
, 1996
"... 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 m ..."
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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 3planar 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...
Programming in Propositional Logic or Reductions: Back to the Roots (Satisfiability)
, 1993
"... In this paper, NPcomplete 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 (nonrandom) f ..."
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Cited by 5 (0 self)
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In this paper, NPcomplete 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 (nonrandom) formulas for SATISFIABILITYsolving 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 : : : : : : : : : : : : : : : : : :...
Simplified O(n) Planarity Algorithms
, 2001
"... 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 pape ..."
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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). In July 1999
Efficient Drawing Algorithms on the Minimum Area for TreeStructured Diagrams
"... In this paper, we deal with a treelike diagram which we call a”tree structured $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{m}$ ” $(TSD$ for short). A $TSD $ is a generalization of program diagrams. We firstly define the problem of drawing TSDs and introduce constraints for b ..."
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In this paper, we deal with a treelike diagram which we call a”tree structured $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{m}$ ” $(TSD$ for short). A $TSD $ is a generalization of program diagrams. We firstly define the problem of drawing TSDs and introduce constraints for beautiful drawings of TSDS. Then we present efficient $O(n)$ and $O(n^{2}) $ algorithms which produces minimum width drawing unders certain sets of constraints. These algorithms will be applied to practical uses such as visual programming and others. 1
Art Galleries With Interior Walls
, 1997
"... . Consider an art gallery formed by a polygon on n vertices with m pairs of vertices joined by interior diagonals, the interior walls. Each interior wall has an arbitrarily placed, arbitrarily small doorway. We will show that the minimum number of guards that suffice to guard all art galleries with ..."
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. Consider an art gallery formed by a polygon on n vertices with m pairs of vertices joined by interior diagonals, the interior walls. Each interior wall has an arbitrarily placed, arbitrarily small doorway. We will show that the minimum number of guards that suffice to guard all art galleries with n vertices and m interior walls is minfb(2n \Gamma 3)=3c; b(2n+m \Gamma 2)=4c; b(2m+n)=3cg. If we restrict ourselves to galleries with convex rooms of size at least r, the answer improves to min fm; b(n + m)=rcg. The proofs lead to linear time algorithms in most cases. The original art gallery problem, posed by Klee and solved by Chv'atal [6], is to find the smallest number of guards necessary to cover any simple polygon, the art gallery, not necessarily convex, on n vertices. Here a covering by g guards means that one can find g points in the interior of the polygon such that every point in the interior is covered by some guard, that is for each point in the interior the line segment betwe...