Parameterized Counting Algorithms for General Graph Covering Problems

Abstract

We examine the general problem of covering graphs by graphs: given a graph G, a collection $\mathcal{P}$ of graphs each on at most p vertices, and an integer r, is there a collection $\mathcal{C}$ of subgraphs of G, each belonging to $\mathcal{P}$, such that the removal of the graphs in $\mathcal{C}$ from G creates a graph none of whose components have more than r vertices? We can also require that the graphs in $\mathcal{C}$ be disjoint (forming a “matching”). This framework generalizes vertex cover, edge dominating set, and minimal maximum matching. In this paper, we examine the parameterized complexity of the counting version of the above general problem. In particular, we show how to count the solutions of size at most k of the covering and matching problems in time O(n · r(pk+r)+2^f(k,p,r)), where n is the number of vertices in G and f is a simple polynomial. In order to achieve the additive relation between the polynomial and the non-polynomial parts of the time complexity of our algorithms, we use the compactor technique, the counting analogue of kernelization for parameterized decision problems.

The two first authors were supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The third author was supported by the Spanish CICYT project TIN-2004-07925 (GRAMMARS).