Simpler Linear-Time Modular Decomposition Via Recursive Factorizing Permutations

Tedder, Marc; Corneil, Derek G.; Habib, Michel; Paul, Christophe

doi:10.1007/978-3-540-70575-8_52

Cited by 121 publications

(107 citation statements)

References 22 publications

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“…Thus it does not matter we use the module or any vertex in it, that is, every vertex is a full representative for the simple series module it lies in. Although there has been a long list of linear algorithms for finding modular decomposition for an undirected graph (see a comprehensive survey by de Montgolfier [23]), it is very time-comsuming because the big constant hidden behind the big-O [26], and considering that the modular decopmosition needs to be re-constructed after each iteration, this will be helpful. It is somehow surprising that the previous kernelization algorithms can be significantly simplified by avoiding modular decomposition.…”

Section: Discussionmentioning

confidence: 99%

Cluster Editing: Kernelization Based on Edge Cuts

Cao

Chen

2011

Algorithmica

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Kernelization algorithms for the cluster editing problem have been a popular topic in the recent research in parameterized computation. Thus far most kernelization algorithms for this problem are based on the concept of critical cliques. In this paper, we present new observations and new techniques for the study of kernelization algorithms for the cluster editing problem. Our techniques are based on the study of the relationship between cluster editing and graph edge-cuts. As an application, we present an O(n 2 )-time algorithm that constructs a 2k kernel for the weighted version of the cluster editing problem. Our result meets the best kernel size for the unweighted version for the cluster editing problem, and significantly improves the previous best kernel of quadratic size for the weighted version of the problem.

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Section: Discussionmentioning

confidence: 99%

Cluster Editing: Kernelization Based on Edge Cuts

Cao

Chen

2011

Algorithmica

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“…[35]). To keep our planarization algorithm simple we will only apply the rule exhaustively on vertices of degree at most 12 + k. This can be done as follows; we pick all of the vertices in G of degree at most 12 + k and sort them by their neighborhoods.…”

Section: Large Matchings In Twin-reduced Graphsmentioning

confidence: 99%

A Near-Optimal Planarization Algorithm

Jansen¹,

Lokshtanov²,

Saurabh³

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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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 n-vertex graph can be made planar by at most k vertex deletions, for every fixed k. Of the known algorithms for k-Vertex Planarization, the algorithm of Marx and Schlotter (WG 2007, Algorithmica 2012) running in time 22 achieves the best running time dependence on k. The algorithm ofthat is not stated explicitly, achieves the best dependence on n.In this paper we present an algorithm for k-Vertex Planarization with running time 2 O(k log k) · n, significantly improving the running time dependence on k without compromising the linear dependence on n. Our main technical contribution is a novel scheme to reduce the treewidth of the input graph to O(k) in time 2 O(k log k) · n. It combines new insights into the structure of graphs that become planar after contracting a matching, with a Baker-type subroutine that reduces the number of disjoint paths through planar parts of the graph that are not affected by the sought solution. To solve the reduced instances we formulate a dynamic programming algorithm for Weighted Vertex Planarization on graphs of treewidth w with running time 2 O(w log w) · n, thereby improving over previous double-exponential algorithms.While Kawarabayashi's planarization algorithm relies heavily on deep results from the graph minors project, our techniques are elementary and practically self-contained. We expect them to be applicable to related edge-deletion and contraction variants of planarization problems.

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“…Moreover, it is not difficult to see that for these graphs the space needed by the algorithm Spanning Trees-Number is O(n + m); recall that the modular decomposition tree of a graph and its construction require space linear in the size of the graph [15,25,34]. Thus, the results of this section are summarized in the following theorem.…”

Section: Counting Spanning Trees In Linear Timementioning

confidence: 94%

“…It is well known that for any graph G the tree T (G) is unique up to isomorphism and it can be constructed in linear time [15,25,34]. A graph is called cograph if it has no induced path on four vertices.…”

Section: Modular Decompositionmentioning

confidence: 99%

“…In particular, for certain classes of graphs, the structure of their modular decomposition trees (and in fact their prime graphs) ensures that each tree node can be processed in time linear in the size of the contracted part of the tree; thus, since the modular decomposition tree of a graph can be constructed in time and space linear in the size of the graph [15,25,34], the processing of the entire modular decomposition tree and consequently the number of its spanning trees takes time and space linear in the size of the input graph. Such classes are the classes of treecographs, (q, q − 4)-graphs for fixed q, and P 4 -tidy graphs, along with their numerous subclasses, such as, the cographs, the P 4 -reducible, the extended P 4 -reducible, the P 4 -sparse, the P 4 -lite, the P 4 -extendible, and the extended P 4 -sparse graphs.…”

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confidence: 99%

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Counting spanning trees using modular decomposition

Nikolopoulos

Palios

Papadopoulos

2014

Theoretical Computer Science

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In this paper we present an algorithm for determining the number of spanning trees of a graph G which takes advantage of the structure of the modular decomposition tree of G. Specifically, our algorithm works by contracting the modular decomposition tree of the input graph G in a bottom-up fashion until it becomes a single node; then, the number of spanning trees of G is computed as the product of a collection of values which are associated with the vertices of G and are updated during the contraction process. In particular, when applied on a (q, q − 4)-graph for fixed q, a P 4 -tidy graph, or a tree-cograph, our algorithm computes the number of its spanning trees in time linear in the size of the graph, where the complexity of arithmetic operations is measured under the uniform-cost criterion. Therefore we give the first linear-time algorithm for the counting problem in the considered graph classes.

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Simpler Linear-Time Modular Decomposition Via Recursive Factorizing Permutations

Cited by 121 publications

References 22 publications

Cluster Editing: Kernelization Based on Edge Cuts

Cluster Editing: Kernelization Based on Edge Cuts

A Near-Optimal Planarization Algorithm

Counting spanning trees using modular decomposition

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