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A Shorter Proof of the Graph Minor Algorithm  The Unique Linkage Theorem
"... At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful theorem which describes the structure of graphs excluding a fixed minor. This result is used to prove Wagner’s conjecture and provide a polynomial time algorithm for the disjoint paths problem when the number of the ..."
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Cited by 17 (6 self)
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At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful theorem which describes the structure of graphs excluding a fixed minor. This result is used to prove Wagner’s conjecture and provide a polynomial time algorithm for the disjoint paths problem when the number of the terminals is fixed (i.e, the Graph Minor Algorithm). However, both results require the full power of the Graph Minor Theory, i.e, the structure theorem. In this paper, we show that this is not true in the latter case. Namely, we provide a new and much simpler proof of the correctness of the Graph Minor Algorithm. Specifically, we prove the “Unique Linkage Theorem ” without using Graph Minors structure theorem. The new argument, in addition to being simpler, is much shorter, cutting the proof by at least 200 pages. We also give a new full proof of correctness of an algorithm for the wellknown edgedisjoint paths problem when the number of the terminals is fixed, which is at most 25 pages long.
Obtaining a planar graph by vertex deletion
 In Proceedings of the 33rd International Workshop on GraphTheoretic Concepts in Computer Science (WG’07). LNCS Series
"... Abstract. In the kApex problem the task is to find at most k vertices whose deletion makes the given graph planar. The graphs for which there exists a solution form a minor closed class of graphs, hence by the deep results of Robertson and Seymour [31, 30], there is an O(n 3) time algorithm for eve ..."
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Cited by 16 (4 self)
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Abstract. In the kApex problem the task is to find at most k vertices whose deletion makes the given graph planar. The graphs for which there exists a solution form a minor closed class of graphs, hence by the deep results of Robertson and Seymour [31, 30], there is an O(n 3) time algorithm for every fixed value of k. However, the proof is extremely complicated and the constants hidden by the bigO notation are huge. Here we give a much simpler algorithm for this problem with quadratic running time, by iteratively reducing the input graph and then applying techniques for graphs of bounded treewidth.
Linear kernels and singleexponential algorithms via protrusion decompositions.
 In Proc. of the 40th International Colloquium on Automata, Languages and Programming (ICALP),
, 2013
"... Abstract A ttreewidthmodulator of a graph G is a set X ⊆ V (G) such that the treewidth of G − X is at most t − 1. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs G that come equipped with a ttreewidthmodulator. Similar decompositions have already been ex ..."
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Cited by 15 (4 self)
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Abstract A ttreewidthmodulator of a graph G is a set X ⊆ V (G) such that the treewidth of G − X is at most t − 1. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs G that come equipped with a ttreewidthmodulator. Similar decompositions have already been explicitly or implicitly used for obtaining polynomial kernels Our first result is that any parameterized graph problem (with parameter k) that has finite integer index and is treewidthbounding admits a linear kernel on the class of Htopologicalminorfree graphs, where H is some arbitrary but fixed graph. A parameterized graph problem is called treewidthbounding if all positive instances have a ttreewidthmodulator of size O(k), for some constant t. This result partially extends previous metatheorems on the existence of linear kernels on graphs of bounded genus Our second application concerns the PlanarFDeletion problem. Let F be a fixed finite family of graphs containing at least one planar graph. Given an nvertex graph G and a nonnegative integer k, PlanarFDeletion asks whether G has a set X ⊆ V (G) such that X k and G − X is Hminorfree for every H ∈ F. This problem encompasses a number of wellstudied parameterized problems such as Vertex Cover, Feedback Vertex Set, and Treewidtht Vertex Deletion. Very recently, an algorithm for PlanarFDeletion with running time 2 O(k) · n log 2 n (such an algorithm is called singleexponential) has been presented in
Obtaining a bipartite graph by contracting few edges
, 2011
"... We initiate the study of the BIPARTITE CONTRACTION problem from the perspective of parameterized complexity. In this problem we are given a graph G on n vertices and an integer k, and the task is to determine whether we can obtain a bipartite graph from G by a sequence of at most k edge contractions ..."
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Cited by 8 (3 self)
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We initiate the study of the BIPARTITE CONTRACTION problem from the perspective of parameterized complexity. In this problem we are given a graph G on n vertices and an integer k, and the task is to determine whether we can obtain a bipartite graph from G by a sequence of at most k edge contractions. Our main result is an f (k) n O(1) time algorithm for BIPARTITE CONTRACTION. Despite a strong resemblance between BIPARTITE CONTRACTION and the classical ODD CYCLE TRANSVERSAL (OCT) problem, the methods developed to tackle OCT do not seem to be directly applicable to BIPARTITE CONTRACTION. To obtain our result, we combine several techniques and concepts that are central in parameterized complexity: iterative compression, irrelevant vertex, and important separators. To the best of our knowledge, this is the first time the irrelevant vertex technique and the concept of important separators are applied in unison. Furthermore, our algorithm may serve as a comprehensible example of the usage of the irrelevant vertex technique.
Interval Deletion is FixedParameter Tractable
, 2014
"... We study the minimum interval deletion problem, which asks for the removal of a set of at most k vertices to make a graph on n vertices into an interval graph. We present a parameterized algorithm of runtime 10k · nO(1) for this problem, thereby showing its fixedparameter tractability. ..."
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Cited by 8 (4 self)
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We study the minimum interval deletion problem, which asks for the removal of a set of at most k vertices to make a graph on n vertices into an interval graph. We present a parameterized algorithm of runtime 10k · nO(1) for this problem, thereby showing its fixedparameter tractability.
Sparse spanners vs. compact routing.
 In Proc. 23th ACM Symp. on Parallel Algorithms and Architectures (SPAA),
, 2011
"... ABSTRACT Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables ofΘ( ..."
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Cited by 5 (2 self)
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ABSTRACT Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables ofΘ(
Obtaining Planarity by Contracting Few Edges
 In Proceedings MFCS 2012, volume 7464 of LNCS
, 2012
"... Abstract. The Planar Contraction problem is to test whether a given graph can be made planar by using at most k edge contractions. This problem is known to be NPcomplete. We show that it is fixedparameter tractable when parameterized by k. 1 ..."
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Abstract. The Planar Contraction problem is to test whether a given graph can be made planar by using at most k edge contractions. This problem is known to be NPcomplete. We show that it is fixedparameter tractable when parameterized by k. 1
A NearOptimal Planarization Algorithm
, 2013
"... The problem of testing whether a graph is planar has been studied for over half a century, and is known to be solvable in O(n) time using a myriad of different approaches and techniques. Robertson and Seymour established the existence of a cubic algorithm for the more general problem of deciding whe ..."
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The problem of testing whether a graph is planar has been studied for over half a century, and is known to be solvable in O(n) time using a myriad of different approaches and techniques. Robertson and Seymour established the existence of a cubic algorithm for the more general problem of deciding whether an nvertex graph can be made planar by at most k vertex deletions, for every fixed k. Of the known algorithms for kVertex Planarization, the algorithm of Marx and Schlotter (WG 2007, Algorithmica 2012) running in time 2k O(k3) · n2 achieves the best running time dependence on k. The algorithm of Kawarabayashi (FOCS 2009), running in time f(k)n for some f(k) ∈ Ω
Planar Induced Subgraphs of Sparse Graphs
, 2015
"... Abstract We show that every graph has an induced pseudoforest of at least n − m/4.5 vertices, an induced partial 2tree of at least n − m/5 vertices, and an induced planar subgraph of at least n−m/5.2174 vertices. These results are constructive, implying lineartime algorithms to find the respectiv ..."
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Abstract We show that every graph has an induced pseudoforest of at least n − m/4.5 vertices, an induced partial 2tree of at least n − m/5 vertices, and an induced planar subgraph of at least n−m/5.2174 vertices. These results are constructive, implying lineartime algorithms to find the respective induced subgraphs. We also show that the size of the largest K h minorfree graph in a given graph can sometimes be at most n − m/6 + o(m).