Parameterized Algorithms for Generalized Domination

Abstract

We study the parameterized complexity of a generalization of Dominating Set problem, namely, the Vector Dominating Set problem. Here, given an undirected graph G = (V,E), with V = {v ₁, ⋯ , v _n }, a vector $\vec{l}=(l(v_1),\cdots, l(v_n))$ and an integer parameter k, the goal is to determine whether there exists a subset D of at most k vertices such that for every vertex v ∈ V ∖ D, at least l(v) of its neighbors are in D. This problem encompasses the well studied problems – Vertex Cover (when l(v) = d(v) for all v ∈ V, where d(v) is the degree of vertex v) and Dominating Set (when l(v) = 1 for all v ∈ V). While Vertex Cover is known to be fixed parameter tractable, Dominating Set is known to be W[2]-complete. In this paper, we identify vectors based on several measures for which this generalized problem is fixed parameter tractable and W-hard. We also show that the Vector Dominating Set is fixed parameter tractable for graphs of bounded degeneracy and for graphs excluding cycles of length four.

The work was done when the second and the third authors were at The Institute of Mathematical Sciences.