We study a generalization of covering problems called partial covering. Here we wish to cover only a desired number of elements, rather than covering all elements as in standard covering problems. For example, in k-partial set cover, we wish to choose a minimum number of sets to cover at least k elements. For k-partial set cover, if each element occurs in at most f sets, then we derive a primal...

Fomin and Villanger ([14], STACS 2010) proved that Maximum Independent Set, Feedback Vertex Set, and more generally the problem of finding a maximum induced subgraph of treewith at most a constant t, can be solved in polynomial time on graph classes with polynomially many minimal separators. We extend these results in two directions. Let Gpoly be the class of graphs with at most poly(n) minimal...

Meta-kernelization theorems are general results that provide polynomial kernels for large classes of parameterized problems. The known meta-kernelization theorems, in particular the results of Bodlaender et al. (FOCS’09) and of Fomin et al. (FOCS’10), apply to optimization problems parameterized by solution size. We present meta-kernelization theorems that use a structural parameters of the inp...

In the SPLIT VERTEX DELETION problem, given a graph G and an integer k, we ask whether one can delete k vertices from the graph G to obtain a split graph (i.e., a graph, whose vertex set can be partitioned into two sets: one inducing a clique and the second one inducing an independent set). In this paper we study fixed-parameter algorithms for SPLIT VERTEX DELETION parameterized by k: we show t...

The standard parameterization of the VERTEX COVER problem (Given an undirected graph G and k ∈ N as input, does G have a vertex cover of size at most k?) has the solution size k as the parameter. The following more challenging parameterization of VERTEX COVER stems from the observation that the size MM of a maximum matching of G lower-bounds the size of any vertex cover of G: Does G have a vert...

Connected Vertex Cover Problem (CVC) is an NP -hard problem. The currently best known approximation algorithm for CVC has performance ration 2. This paper gives the first Polynomial Time Approximation Scheme for CVC in Unit Disk Graph.

TheOdd Cycle Transversal problem (oct) asks whether a given graph can be made bipartite (i.e., 2-colorable) by deleting at most l vertices. We study structural parameterizations of oct with respect to their polynomial kernelizability, i.e., whether instances can be efficiently reduced to a size polynomial in the chosen parameter. It is a major open problem in parameterized complexity whether Od...

DOMINATING SET remains NP -complete even when instances are restricted to bipartite graphs, however, in this case VERTEX COVER is solvable in polynomial time. Consequences to VECTOR DOMINATING SET as a generalization of both are discussed.

We give a necessary and sufficient condition for the maximum multiplicity of a root of the matching polynomial of a tree to be equal to the minimum number of vertex disjoint paths needed to cover it.

Capacitated versions of Vertex Cover and Dominating Set have been studied intensively in terms of polynomial time approximation algorithms. Although the problems Dominating Set and Vertex Cover have been subjected to considerable scrutiny in the parameterized complexity world, this is not true for their capacitated versions. Here we make an attempt to understand the behavior of the problems Cap...