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 k-signed r-set on [n] = {1, ..., n} is an ordered pair (A, f), where A is an rsubset of [n] and f is a function from A to [k]. Families A1, ...,Ap are said to be cross-intersecting if any set in any family Ai intersects any set in any other family Aj . Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of [n]. Our aim is to generalise Hilton's bound t...

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...

A Smarandachely k-signed graph (Smarandachely k-marked graph) is an ordered pair S = (G,σ) (S = (G,μ)) where G = (V, E) is a graph called underlying graph of S and σ : E → (e1, e2, ..., ek) (μ : V → (e1, e2, ..., ek)) is a function, where each ei ∈ {+,−}. Particularly, a Smarandachely 2-signed graph or Smarandachely 2-marked graph is called abbreviated a signed graph or a marked graph. Given a ...

For an integer k ≥ 1 and a graph G = (V,E), a set S of V is k-independent if ∆(S) < k and k-dominating if every vertex in V \S has at least k neighbors in S. The k-independence number βk(G) is the maximum cardinality of a k-independent set and the k-dominating number is the minimum cardinality of a k-dominating set of G. Since every kindependent set is (k + 1)-independent and every (k + 1)-domi...