Learn More
Abstract. We present an algorithm that constructively produces a solution to the k -DOMINATING SET problem for planar graphs in time O(c^ \sqrt k n) , where c=4^ 6\sqrt 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ (G) is O(\sqrt \rule 0pt 4pt \smash γ (G) ) , and that such a tree decomposition can be(More)
In this paper, we give evidence for the problems Disjoint Cycles and Disjoint Paths that they cannot be preprocessed in polynomial time such that resulting instances always have a size bounded by a polynomial in a specified parameter (or, in short: do not have a polynomial kernel); these results are assuming the validity of certain complexity theoretic(More)
It is shown that the problem of maintaining the topological order of the nodes of a directed acyclic graph while inserting <i>m</i> edges can be solved in <i>O</i>(min{<i>m</i><sup>3/2 log <i>n m</i>3/3</sup> + <i>n</i><sup>2</sup> log <i>n</i>}) time, an improvement over the best known result of <i>O</i>(<i>mn</i>). In addition, we analyze the complexity(More)
There are many graph problems that can be solved in linear or polynomial time with a dynamic programming algorithm when the input graph has bounded treewidth. For combinatorial optimization problems, this is a useful approach for obtaining fixed-parameter tractable algorithms. Starting from trees and series-parallel graphs, we introduce the concepts of(More)
We introduce the cross-composition framework for proving kernelization lower bounds. A classical problem L and/or-cross-composes into a parameterized problem Q if it is possible to efficiently construct an instance of Q with polynomially bounded parameter value that expresses the logical and or or of a sequence of instances of L. Building on work by(More)