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It’s in the Bag: Plane Decompositions as Tools for Approximation
, 2005
"... 2. Background............................... 8 3. Tree Decompositions.......................... 13 ..."
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2. Background............................... 8 3. Tree Decompositions.......................... 13
Approximation algorithms for the Maximum Induced Planar and Outerplanar Subgraph problems
 J. Graph Algorithms Appl
"... The task of finding the largest subset of vertices of a graph that induces a planar subgraph is known as the Maximum Induced Planar Subgraph problem (MIPS). In this paper, some new approximation algorithms for MIPS are introduced. The results of an extensive study of the performance of these and exi ..."
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The task of finding the largest subset of vertices of a graph that induces a planar subgraph is known as the Maximum Induced Planar Subgraph problem (MIPS). In this paper, some new approximation algorithms for MIPS are introduced. The results of an extensive study of the performance of these and existing MIPS approximation algorithms on randomly generated graphs are presented. Efficient algorithms for finding large induced outerplanar graphs are also given. One of these algorithms is shown to find an induced outerplanar subgraph with at least 3n/(d + 5/3) vertices. The results presented in this paper indicate that most existing algorithms perform substantially better than the existing lower bounds indicate. 1
A planarity test via construction sequences
 CoRR
"... Abstract. Lineartime algorithms for testing the planarity of a graph are well known for over 35 years. However, these algorithms are quite involved and recent publications still try to give simpler lineartime tests. We give a conceptually simple reduction from planarity testing to the problem of c ..."
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Abstract. Lineartime algorithms for testing the planarity of a graph are well known for over 35 years. However, these algorithms are quite involved and recent publications still try to give simpler lineartime tests. We give a conceptually simple reduction from planarity testing to the problem of computing a certain construction of a 3connected graph. This implies a lineartime planarity test. Our approach is radically different from all previous lineartime planarity tests; as key concept, we maintain a planar embedding that is 3connected at each point in time. The algorithm computes a planar embedding if the input graph is planar and a Kuratowskisubdivision otherwise. 1
Two New Approximation Algorithms for the Maximum Planar Subgraph Problem
, 2006
"... The maximum planar subgraph problem (MPS) is defined as follows: given a graph G, find a largest planar subgraph of G. The problem is NPhard and it has applications in graph drawing and resource location optimization. Călinescu et al. [J. Alg. 27, 269302 (1998)] presented the first approximation a ..."
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The maximum planar subgraph problem (MPS) is defined as follows: given a graph G, find a largest planar subgraph of G. The problem is NPhard and it has applications in graph drawing and resource location optimization. Călinescu et al. [J. Alg. 27, 269302 (1998)] presented the first approximation algorithms for MPS with nontrivial performance ratios. Two algorithms were given, a simple algorithm which runs in linear time for boundeddegree graphs with a ratio 7/18 and a more complicated algorithm with a ratio 4/9. Both algorithms produce outerplanar subgraphs. In this article we present two new versions of the simpler algorithm. The first new algorithm still runs in the same time, produces outerplanar subgraphs, has at least the same performance ratio as the original algorithm, but in practice it finds larger planar subgraphs than the original algorithm. The second new algorithm has similar properties to the first algorithm, but it produces only planar subgraphs. We conjecture that the performance ratios of our algorithms are at least 4/9 for MPS. We experimentally compare the new algorithms against the original simple algorithm. We also apply the new algorithms for approximating the thickness and outerthickness of a graph. Experiments show that the new algorithms produce clearly better approximations than the original simple algorithm by Călinescu et al.
Optimal NodeDegree Bounds for the Complexity of Nonplanarity Parameters
 IN PROC. TENTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS, SODA'99
, 1999
"... We prove that both the NPcompleteness of the nonplanar deletion decision problem and the Max SNPhardness of the nonplanar deletion problem remain true even for cubic graphs. We prove that the class of graphs with splitting number less than or equal to a fixed k is minor closed, which implies the e ..."
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We prove that both the NPcompleteness of the nonplanar deletion decision problem and the Max SNPhardness of the nonplanar deletion problem remain true even for cubic graphs. We prove that the class of graphs with splitting number less than or equal to a fixed k is minor closed, which implies the existence of a corresponding polynomialtime recognition algorithm.