Determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms. Theoretically, it has been proved that a parameterized problem is kernel-izable if and only if it is fixed-parameter tractable. Practically,… (More)
We discuss general techniques, centered around the " Layerwise Separation Property " (LSP) of a planar graph problem, that allow to develop algorithms with running time c √ k |G|, given an instance G of a problem on planar graphs with parameter k. Problems having LSP include planar vertex cover, planar independent set, and planar dominating set. Extensions… (More)
In this paper, we show how to systematically improve on parame-terized algorithms and their analysis, focusing on search-tree based algorithms for d-Hitting Set, especially for d = 3. We concentrate on algorithms which are easy to implement, in contrast with the highly sophisticated algorithms which have been elsewhere designed to improve on the exponential… (More)
The notion of a " problem kernel " plays a central role in the design of fixed-parameter algorithms. The FPT literature is rich in kernelization algorithms that exhibit fundamentally different approaches. We highlight these differences and discuss several generalizations and restrictions of the standard notion.
The <i>k</i>-Leaf Out-Branching problem is to find an out-branching, that is a rooted oriented spanning tree, with at least <i>k</i> leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the <i>k</i>-Leaf-Out-Branching problem. We give the… (More)
We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c √ k n), where c = 3 6 √ 34. To obtain this result, we show that the treewidth of a pla-nar graph with domination number γ(G) is O(γ(G)), and that such a tree decomposition can be found in O(γ(G)n) time. The same technique can be used… (More)
Two trees with the same number of leaves have to be embedded in two layers in the plane such that the leaves are aligned in two adjacent layers. Additional matching edges between the leaves give a one-to-one correspondence between pairs of leaves of the different trees. Do there exist two planar embeddings of the two trees that minimize the crossings of the… (More)
We consider the concepts of a t-total vertex cover and a t-total edge cover (t ≥ 1), which generalize the notions of a vertex cover and an edge cover, respectively. A t-total vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of the subgraph of G induced by S has least t vertices… (More)