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Efficient Algorithms for Drawing Planar Graphs
, 1999
"... 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 ................. ..."
Abstract

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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 Boxrectangular 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...
Approximation and Inaproximability Results on Balanced Connected Partitions of Graphs
"... Let G = (V, E) be a connected graph with a weight function w: V → Z+ and let q ≥ 2 be a positive integer. For X ⊆ V, let w(X) denote the sum of the weights of the vertices in X. We consider the following problem on G: find a qpartition P = (V1, V2,..., Vq) of V such that G[Vi] is connected (1 ≤ i ≤ ..."
Abstract
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Let G = (V, E) be a connected graph with a weight function w: V → Z+ and let q ≥ 2 be a positive integer. For X ⊆ V, let w(X) denote the sum of the weights of the vertices in X. We consider the following problem on G: find a qpartition P = (V1, V2,..., Vq) of V such that G[Vi] is connected (1 ≤ i ≤ q) and P maximizes min{w(Vi) : 1 ≤ i ≤ q}. This problem is called Max Balanced Connected qPartition and is denoted by BCPq. We show that for q ≥ 2 the problem BCPq is NPhard in the strong sense, even on qconnected graphs, and therefore does not admit a FPTAS, unless P = NP. We also show another inapproximability result for BCP2. For the problemapproximation algorithm obtained by Chlebíková; for q = 3 and q = 4 we present 2approximation algorithms. When q is not fixed (it is part of the instance), the corresponding problem is called Max Balanced Connected Partition, and denoted as BCP. We show that BCP does not admit an approximation algorithm with ratio smaller than 6/5, unless P = NP. BCPq restricted to qconnected graphs, it is known that for q = 2 the best result is a 4 3