In this paper we study the NP-complete problem of finding small k-dominating sets in general graphs, which allow k−1 nodes to fail and still dominate the graph. The classic minimum dominating set problem is a special case with k = 1. We show that the approach of having at least k dominating nodes in the neighborhood of every node is not optimal. For each α > 1 it can give solutions k α times la...

A graph is k-connected if it has k internally-disjoint paths between every pair of nodes. A subset S of nodes in a graph G is a kconnected set if the subgraph G[S] induced by S is k-connected; S is an m-dominating set if every v ∈ V \ S has at least m neighbors in S. If S is both k-connected and m-dominating then S is a k-connected m-dominating set, or (k,m)-cds for short. In the k-Connected mD...

The top-k dominating query returns the k database objects with the highest score with respect to their dominance score. The dominance score of an object p is simply the number of objects dominated by p, based on minimization or maximization preferences on the attribute values. Each object (tuple) is represented as a point in a multidimensional space, and therefore, the number of attributes equa...

We show that the problem k-DOMINATING SET and its several variants including k-CONNECTED DOMINATING SET, k-INDEPENDENT DOMINATING SET, and k-DOMINATING CLIQUE, when parameterized by the solution size k, are W[1]-hard in either multiple-interval graphs or their complements or both. On the other hand, we show that these problems belong to W[1] when restricted to multipleinterval graphs and their ...

Let G be a graph with vertex set V (G), and let f : V (G) −→ {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑ x∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of distinct signed total k-dominating functions on G with the property that ∑d i=1 fi(x) ≤ k for each x ∈ V (G), is call...

Many NP-hard problems, such as Dominating Set, are FPT parameterized by clique-width. For graphs of clique-width k given with a kexpression, Dominating Set can be solved in 4knO(1) time. However, no FPT algorithm is known for computing an optimal k-expression. For a graph of clique-width k, if we rely on known algorithms to compute a (23k − 1)expression via rank-width and then solving Dominatin...

Many NP-hard problems, such as Dominating Set, are FPT parameterized by clique-width. For graphs of clique-width k given with a kexpression, Dominating Set can be solved in 4knO(1) time. However, no FPT algorithm is known for computing an optimal k-expression. For a graph of clique-width k, if we rely on known algorithms to compute a (23k − 1)expression via rank-width and then solving Dominatin...

Perhaps the best known kernelization result is the kernel of size 335k for the Planar Dominating Set problem by Alber et al. [1], later improved to 67k by Chen et al. [5]. This result means roughly, that the problem of finding the smallest dominating set in a planar graph is easy when the optimal solution is small. On the other hand, it is known that Planar Dominating Set parameterized by k = |...

Let D be a simple digraph with vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑x∈N−[v] f(x) ≥ k for each v ∈ V (D), where N[v] consists of v and all vertices of D from which arcs go into v, then f is a signed k-dominating function on D. A set {f1, f2, . . . , fd} of distinct signed k-dominating functions on D with the property that ∑d i=1 fi(x...