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Experiments on exact crossing minimization using column generation
 ACM Journal of Experimental Algorithmics
"... Abstract. The crossing number of a graph G is the smallest number of edge crossings in any drawing of G into the plane. Recently, the first branchandcut approach for solving the crossing number problem has been presented in [3]. Its major drawback was the huge number of variables out of which only ..."
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Abstract. The crossing number of a graph G is the smallest number of edge crossings in any drawing of G into the plane. Recently, the first branchandcut approach for solving the crossing number problem has been presented in [3]. Its major drawback was the huge number of variables out of which only very few were actually used in the optimal solution. This restricted the algorithm to rather small graphs with low crossing number. In this paper we discuss two column generation schemes; the first is based on traditional algebraic pricing, and the second uses combinatorial arguments to decide whether and which variables need to be added. The main focus of this paper is the experimental comparison between the original approach, and these two schemes. We also compare these new results to the solutions of the best known crossing number heuristic. 1
A linear time algorithm for finding a maximal planar subgraph based on PCtrees
 of Lecture Notes in Computer Science
, 2005
"... ABSTRACT. Given an undirected graph G, the maximal planar subgraph problem is to determine a planar subgraph H of G such that no edge of GH can be added to H without destroying planarity. Polynomial algorithms have been obtained by Jakayumar, Thulasiraman and Swamy [6] and Wu [9]. O(mlogn) algorith ..."
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ABSTRACT. Given an undirected graph G, the maximal planar subgraph problem is to determine a planar subgraph H of G such that no edge of GH can be added to H without destroying planarity. Polynomial algorithms have been obtained by Jakayumar, Thulasiraman and Swamy [6] and Wu [9]. O(mlogn) algorithms were previously given by Di Battista and Tamassia [3] and Cai, Han and Tarjan [2]. A recent O(mα (n)) algorithm was obtained by La Poute [7]. Our algorithm is based on a simple planarity test [5] developed by the author, which is a vertex addition algorithm based on a depthfirstsearch ordering. The planarity test [5] uses no complicated data structure and is conceptually simpler than Hopcroft and Tarjan's path addition and Lempel, Even and Cederbaum's vertex addition approaches. 1 1.
Embedding a Graph Into the Torus in Linear Time
, 1994
"... A linear time algorithm is presented that, for a given graph G, finds an embedding of G in the torus whenever such an embedding exists, or exhibits a subgraph\Omega of G of small branch size that cannot be embedded in the torus. 1 Introduction Let K be a subgraph of G, and suppose that we are ..."
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A linear time algorithm is presented that, for a given graph G, finds an embedding of G in the torus whenever such an embedding exists, or exhibits a subgraph\Omega of G of small branch size that cannot be embedded in the torus. 1 Introduction Let K be a subgraph of G, and suppose that we are given an embedding of K in some surface. The embedding extension problem asks whether it is embedding extension problem possible to extend the embedding of K to an embedding of G in the same surface, and any such embedding is an embedding extension of K to G. An embedding extension obstruction for embedding extensions is a subgraph\Omega of G \Gamma E(K) such that obstruction the embedding of K cannot be extended to K [ \Omega\Gamma The obstruction is small small if K [\Omega is homeomorphic to a graph with a small number of edges. If\Omega is small, then it is easy to verify (for example, by checking all the possibilities Supported in part by the Ministry of Science and Technolo...
An Implementation of the Hopcroft and Tarjan Planarity Test and Embedding Algorithm
, 1993
"... We describe an implementation of the Hopcroft and Tarjan planarity test and embedding algorithm. The program tests the planarity of the input graph and either constructs a combinatorial embedding (if the graph is planar) or exhibits a Kuratowski subgraph (if the graph is nonplanar). ..."
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We describe an implementation of the Hopcroft and Tarjan planarity test and embedding algorithm. The program tests the planarity of the input graph and either constructs a combinatorial embedding (if the graph is planar) or exhibits a Kuratowski subgraph (if the graph is nonplanar).
An efficient implementation of the PCtrees algorithm of shih and hsu’s planarity test
 Institute of Information Science, Academia Sinica
, 2003
"... In Shih & Hsu [9] a simpler planarity test was introduced utilizing a data structure called PCtrees (generalized from PQtrees). In this paper we give an efficient implementation of that linear time algorithm and illustrate in detail how to obtain a Kuratowski subgraph when the given graph is not p ..."
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In Shih & Hsu [9] a simpler planarity test was introduced utilizing a data structure called PCtrees (generalized from PQtrees). In this paper we give an efficient implementation of that linear time algorithm and illustrate in detail how to obtain a Kuratowski subgraph when the given graph is not planar, and how to obtain the embedding alongside the testing algorithm. We have implemented the algorithm using LEDA and an object code is available at
Simpler Projective Plane Embedding
, 2000
"... A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n ..."
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A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n 2 ) time but is much easier to implement. We programmed a variant of this algorithm and used it to computationally verify the known list of all the projective plane obstructions. Key words: graph algorithms, surface embedding, graph embedding, projective plane, forbidden minor, obstruction 1 Background A graph G consists of a set V of vertices and a set E of edges, each of which is associated with an unordered pair of vertices from V . Throughout this paper, n denotes the number of vertices of a graph, and m is the number of edges. A graph is embeddable on a surface M if it can be drawn on M without crossing edges. Archdeacon's survey [2] provides an excellent introduction to topologica...
Planarity In Linear Time
, 1997
"... I give a selfcontained exposition of a lineartime planarity algorithm of Shih and Hsu. ..."
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I give a selfcontained exposition of a lineartime planarity algorithm of Shih and Hsu.
Efficient Extraction of Multiple Kuratowski Subdivisions (TR)
, 2007
"... Abstract. A graph is planar if and only if it does not contain a Kuratowski subdivision. Hence such a subdivision can be used as a witness for nonplanarity. Modern planarity testing algorithms allow to extract a single such witness in linear time. We present the first linear time algorithm which is ..."
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Abstract. A graph is planar if and only if it does not contain a Kuratowski subdivision. Hence such a subdivision can be used as a witness for nonplanarity. Modern planarity testing algorithms allow to extract a single such witness in linear time. We present the first linear time algorithm which is able to extract multiple Kuratowski subdivisions at once. This is of particular interest for, e.g., BranchandCut algorithms which require multiple such subdivisions to generate cut constraints. The algorithm is not only described theoretically, but we also present an experimental study of its implementation. 1
An algorithm for embedding graphs in the torus
"... An efficient algorithm for embedding graphs in the torus is presented. Given a graph G, the algorithm either returns an embedding of G in the torus or a subgraph of G which is a subdivision of a minimal nontoroidal graph. The algorithm based on [13] avoids the most complicated step of [13] by applyi ..."
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An efficient algorithm for embedding graphs in the torus is presented. Given a graph G, the algorithm either returns an embedding of G in the torus or a subgraph of G which is a subdivision of a minimal nontoroidal graph. The algorithm based on [13] avoids the most complicated step of [13] by applying a recent result of Fiedler, Huneke, Richter, and Robertson [5] about the genus of graphs in the projective plane, and simplifies other steps on the expense of losing linear time complexity. 1
Planarity In Linear Time (Class notes)
, 1997
"... I give a selfcontained exposition of a lineartime planarity algorithm of Shih and Hsu. 20 December 1995 Revised 6 June 1997 Partially supported by NSF under Grant No. DMS9303761 and by ONR under Grant No. N0001493 10325. 1. INTRODUCTION There are four lineartime planarity algorithms that ..."
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I give a selfcontained exposition of a lineartime planarity algorithm of Shih and Hsu. 20 December 1995 Revised 6 June 1997 Partially supported by NSF under Grant No. DMS9303761 and by ONR under Grant No. N0001493 10325. 1. INTRODUCTION There are four lineartime planarity algorithms that I am aware of [1, 3, 4, 5, 6], of which the one in [5] (thereafter referred to as the S&H algorithm) seems to be the simplest. The purpose of this note is to give a selfcontained presentation of this algorithm. In section 2 we prove a structural result that is the basis for the correctness of the S&H algorithm, and illustrate it by deducing Kuratowski's theorem from it. The structural result immediately gives a quadratic planarity algorithm. In Section 3 we show how to implement it to run in linear time and linear space for 3connected graphs on a RAM machine. 2. PLANAR GRAPHS By a graph we mean a finite, undirected, simple graph. This is without loss of generality, because a multigraph ...