In this paper, we consider counting the number of ways to place kings on an k × n chessboard, such that every square is dominated by a king. Let f(k, n) be the number of dominating configurations. We consider the asymptotic behavior of the function f(k, n).

We propose a fast silent self-stabilizing building a k-independent dominating set, named FID. The convergence of protocol FID, is established for any computation under the unfair distributed scheduler. FID reaches a terminal (also legitimate) configuration in at most 4n+k rounds, where n is the network size. FID requires (k + 1)log(n+ 1) bits per node. keywords distributed computing, fault tole...

Given a graph G, a k-dominating set of G is a subset S of its vertices with the property that every vertex of G is either in S or has at least k neighbors in S. We present a new incremental local algorithm to construct a k-dominating set. The algorithm constructs a monotone family of dominating sets D1 ⊆ D2 . . . ⊆ Di . . . ⊆ Dk such that each Di is an i-dominating set. For unit disk graphs, th...

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...

For MSO2-expressible problems like Edge Dominating Set or Hamiltonian Cycle, it was open for a long time whether there is an algorithm which given a clique-width k-expression of an n-vertex graph runs in time f(k) · nO(1) for some function f . Recently, Fomin et al. (SIAM. J. Computing, 2014) presented several lower bounds; for instance, there are no f(k) · n-time algorithms for Edge Dominating...

We propose a fast silent self-stabilizing building a k-independent dominating set, named FID. The convergence of protocol FID, is established for any computation under the unfair distributed scheduler. FID reaches a terminal (also legitimate) configuration in at most 4n+k rounds, where n is the network size. FID requires (k + 1)log(n+ 1) bits per node. keywords distributed computing, fault tole...

It is shown that the problem of covering an n x n chessboard with a minimum number of queens on a major diagonal is related to the number-theoretic function rj(n), the smallest number of integers in a subset of {l,..., n} which must contain three terms in arithmetic progression. Several problems concerning the covering of chessboards by queens have been studied in the literature [2]. In this no...

The k-dominating graph Dk(G) of a graph G is defined on the vertex set consisting of dominating sets of G with cardinality at most k, two such sets being adjacent if they differ by either adding or deleting a single vertex. In this paper, after presenting several basic properties of k-dominating graphs, it is proved that if G is a graph with no isolates, of order n ≥ 2, and with G ∼= Dk(G), the...