The base of a signed digraph S is the minimum number k such that for any vertices u, v of S, there is a pair of walks of length k from u to v with different signs. Let K be a signed complete graph of order n, which is a signed digraph obtained by assigning +1 or −1 to each arc of the n-th order complete graph Kn considered as a digraph. In this paper we show that for n ≥ 3 the base of a primiti...

Kamei and Kakugawa have recently proposed a self-stabilizing algorithm for the minimal k-dominating set problem. Their algorithm is a general form of the maximalindependent-set algorithm proposed by Shukla et al. The results in their paper are for any tree network that assumes Dijkstra’s central demon model. In particular, the worstcase stabilization time is claimed to be O(n), where n is the n...

Let k be a positive integer and G = (V,E) be a simple graph. A subset S ⊆ V is dominating in G, if for each vertex v ∈ V \ S, N(v) ∩ S 6= ∅. In 1985, Fink and Jacobson gave a generalization of the concept of dominating sets in graphs. A subset S of V is kdominating in G, if every vertex of V \ S is adjacent to at least k vertices in S. In this paper, we characterize vertices that are in all or ...

Vertex Cover (VC): A vertex cover in an undirected graph G = (V,E) is a subset of vertices V ′ ⊆ V such that every edge in G has at least one endpoint in V ′. The vertex cover problem (VC) is: given an undirected graph G and an integer k, does G have a vertex cover of size k? Dominating Set (DS): A dominating set in a graph G = (V,E) is a subset of vertices V ′ such that every vertex in the gra...

A vertex set D in a graph G is k-dependent if G[D] has maximum degree at most k−1, and k-dominating if every vertex outside D has at least k neighbors in D. Favaron [2] proved that if D is a k-dependent set maximizing the quantity k |D|−|E(G[D])|, then D is k-dominating. We extend this result, showing that such sets satisfy a stronger property: given any ordering < of V (G)−D, there is a k-edge...

We de ne an analog of signed Eulerian numbers fn,k for involutions of the symmetric group and derive some combinatorial properties of this sequence. In particular, we exhibit both an explicit formula and a recurrence for fn,k arising from the properties of its generating function.