We initiate the first systematic study of the NP-hard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixed-parameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. The parameter is the number of vertex deletions. We present efficient fixed-parameter algorithms for CVD applying the fairly new iterative compression technique. Moreover, we study the variant of CVD where the maximum number of cliques to be generated is prespecified. Here, we exploit connections to fixed-parameter algorithms for (weighted) Vertex Cover.

The r-Regular Induced Subgraph problem asks, given a graph G and a non-negative integer k, whether G contains an r-regular induced subgraph of size at least k, that is, an induced subgraph in which every vertex has degree exactly r. In this paper we examine its parameterization k-Size r-Regular Induced Subgraph with k as parameter and prove that it is W [1]-hard. We also examine the parameterized complexity of the dual parameterized problem, namely, the k-Almost r-Regular Graph problem, which asks for a given graph G and a non-negative integer k whether G can be made r-regular by deleting at most k vertices. For this problem, we prove the existence of a problem kernel of size O (kr(r + k) 2 ).

Finding subgraphs of small diameter in undirected graphs has been seemingly unexplored from a parameterized complexity perspective. We perform the first parameterized complexity study on the corresponding NP-hard s-Club problem. We consider two parameters: the solution size and its dual.

No abstract

a b s t r a c tIn an undirected graph G = (V , E), a set of k vertices is called c-isolated if it has less than c · k outgoing edges. Ito and Iwama [H. Ito, K. Iwama, Enumeration of isolated cliques and pseudo-cliques, ACM Transactions on Algorithms (2008) (in press)] gave an algorithm to enumerate all c-isolated maximal cliques in O(4 c · c 4 · |E|) time. We extend this to enumerating all maximal c-isolated cliques (which are a superset) and improve the running time bound to O(2.89 c · c 2 · |E|), using modifications which also facilitate parallelizing the enumeration. Moreover, we introduce a more restricted and a more general isolation concept and show that both lead to faster enumeration algorithms. Finally, we extend our considerations to s-plexes (a relaxation of the clique notion), providing a W[1]-hardness result when the size of the s-plex is the parameter and a fixed-parameter algorithm for enumerating isolated s-plexes when the parameter describes the degree of isolation.

We initiate the first systematic study of the NP-hard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixed-parameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. The parameter is the number of vertex deletions. We present efficient fixed-parameter algorithms for CVD applying the fairly new iterative compression technique. Moreover, we study the variant of CVD where the maximum number of cliques to be generated is prespecified. Here, we exploit connections to fixed-parameter algorithms for (weighted) Vertex Cover.

No abstract

No abstract