A k-dominating set of a graph G is a subset D of the vertices of G such that every vertex of G is either in D or at distance at most k from a vertex in D. It is of interest to find k-dominating sets of small cardinality. In this paper we consider simple randomised greedy algorithms for finding small k-dominating sets of regular graphs. We analyse the average-case performance of the most efficie...

In the Connected Dominating Set problem we are given as input a graph G and a positive integer k, and are asked if there is a set S of at most k vertices of G such that S is a dominating set of G and the subgraph induced by S is connected. This is a basic connectivity problem that is known to be NP-complete, and it has been extensively studied using several algorithmic approaches. In this paper...

Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) −→ {−1, 1} is said to be a signed star k-dominating function on G if ∑ e∈E(v) f(e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. A set {f1, f2, . . . , fd} of signed star k-dominating functions on G with the property that ∑d i=1 fi(e)...

Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) −→ {±1,±2, . . . ,±k} is said to be a signed star {k}-dominating function on G if ∑ e∈E(v) f(e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. The signed star {k}-domination number of a graph G is γ{k}SS(G) = min{ ∑ e∈E f(e) | f is a S...

Phylogenies are useful for organizing knowledge of biological diversity, for structuring classifications, and for providing knowledge of events that occurred during evolution. Different phylogenetic reconstruction techniques are available. In this paper Distance based technique is used. Distance measure is an important issue in phylogenetic analysis. Traditional approaches are time-consuming du...

Let S be a finite set. Let k be a positive integer. Let A be a subset of Sk satisfying |A| = |S|k−1. Let B = Sk \ A. For every v ∈ Sk and every i ∈ {1, 2, . . . , k}, we denote by vi the i-th component of the k-tuple v (remember that v is an element of Sk, that is, a k-tuple of elements of S). Then, every v ∈ Sk satisfies v = (v1, v2, . . . , vk). Let F be the set of all pairs (a, b) ∈ A × B fo...

For a positive integer k, a k-rainbow dominating function of a graph G is a function f from the vertex set V (G) to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V (G) with f(v) = ∅ the condition ⋃ u∈N(v) f(u) = {1, 2, . . . , k} is fulfilled, where N(v) is the neighborhood of v. The 1-rainbow domination is the same as the ordinary domination. A set {f1, f2, ....