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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
Explicit linear kernels via dynamic programming
 IN STACS, VOLUME 25 OF LIPICS
, 2014
"... Several algorithmic metatheorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding ..."
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Cited by 4 (2 self)
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Several algorithmic metatheorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding a fixed topological minor. Typically, these results guarantee the existence of linear or polynomial kernels on sparse graph classes for problems satisfying some generic conditions but, manly due to their generality, it is not known how to derive from them constructive kernels with explicit constants. In this paper we make a step toward a fully constructive metakernelization theory on sparse graphs. Our approach is based on a more explicit protrusion replacement machinery that, instead of expressibility in CMSO logic, uses dynamic programming, which allows us to find an explicit upper bound on the size of the derived kernels. We demonstrate the usefulness of our techniques by providing the first explicit linear kernels for rDominating Set and rScattered Set on apexminorfree graphs, and for PlanarFDeletion and PlanarFPacking on graphs excluding a fixed (topological) minor in the case where all the graphs in F are connected.
Fast Minor Testing in Planar Graphs
, 2012
"... Minor Containment is a fundamental problem in Algorithmic Graph Theory used as a subroutine in numerous graph algorithms. A model of a graph H in a graph G is a set of disjoint connected subgraphs of G indexed by the vertices of H, such that if {u,v} is an edge of H, then there is an edge of G bet ..."
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Cited by 2 (2 self)
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Minor Containment is a fundamental problem in Algorithmic Graph Theory used as a subroutine in numerous graph algorithms. A model of a graph H in a graph G is a set of disjoint connected subgraphs of G indexed by the vertices of H, such that if {u,v} is an edge of H, then there is an edge of G between components Cu and Cv. A graph H is a minor of G if G contains a model of H as a subgraph. We give an algorithm that, given a planar nvertex graph G and an hvertex graph H, either finds in time O(2O(h) ·n+n2 · logn) a model of H in G, or correctly concludes that G does not contain H as a minor. Our algorithm is the first singleexponential algorithm for this problem and improves all previous minor testing algorithms in planar graphs. Our technique is based on a novel approach called partially embedded dynamic programming.
A practical heuristic for finding graph minors
, 2014
"... We present a heuristic algorithm for finding a graph H as a minor of a graph G that is practical for sparse G and H with hundreds of vertices. We also explain the practical importance of finding graph minors in mapping quadratic pseudoboolean optimization problems onto an adiabatic quantum annealer ..."
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Cited by 1 (0 self)
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We present a heuristic algorithm for finding a graph H as a minor of a graph G that is practical for sparse G and H with hundreds of vertices. We also explain the practical importance of finding graph minors in mapping quadratic pseudoboolean optimization problems onto an adiabatic quantum annealer. 1
WellQuasiOrders in Subclasses of Bounded Treewidth Graphs and their Algorithmic Applications
"... We show that three subclasses of bounded treewidth graphs are wellquasiordered by refinements of the minor order. Specifically, we prove that graphs with bounded vertexcovers are well quasi ordered by the induced subgraph order, graphs with bounded feedbackvertexset are well quasi ordered by t ..."
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We show that three subclasses of bounded treewidth graphs are wellquasiordered by refinements of the minor order. Specifically, we prove that graphs with bounded vertexcovers are well quasi ordered by the induced subgraph order, graphs with bounded feedbackvertexset are well quasi ordered by the topologicalminor order, and graphs with bounded circumference are well quasi ordered by the inducedminor order. Our results give algorithms for recognizing any graph family in these classes which is closed under the corresponding minor order refinement.