We propose a memory efficient self-stabilizing protocol building k-independent dominating sets. A k-independent dominating set is a k-independent set and a kdominating set. A set of nodes, I, is k-independent if the distance between any pair of nodes in I is at least k + 1. A set of nodes, D, is a k-dominating if every node is within distance k of a node of D. Our algorithm, named SID, is silen...

Preprocessing by data reduction is a simple but powerful technique used for practically solving different network problems. A number of empirical studies shows that a set of reduction rules for solving Dominating Set problems introduced by Alber, Fellows & Niedermeier leads efficiently to optimal solutions for many realistic networks. Despite of the encouraging experiments, the only class of gr...

In this paper, we are concerned with the krainbow domination problem on generalized de Bruijn digraphs. We give an upper bound and a lower bound for the k-rainbow domination number in generalized de Bruijn digraphs GB(n, d). We also show that γrk(GB(n, d)) = k if and only if α 6 1, where n = d+α and γrk(GB(n, d)) is the k-rainbow domination number of GB(n, d).

A distance-k dominating set S of a directed graph D is a set of vertices such that for every vertex v of D, there is a vertex u ∈ S at distance at most k from it. Minimum distance-k dominating set is especially important in communication networks for distributed data structure and for server placement. In this paper, we shall present simple distributed algorithms for finding the unique minimum ...

A distance-k dominating set S of a directed graph D is a set of vertices such that for every vertex v of D, there is a vertex u ∈ S at distance at most k from it. Minimum distance-k dominating set is especially important in communication networks for distributed data structure and for server placement. In this paper, we shall present simple distributed algorithms for finding the unique minimum ...

Let G be a properly colored bipartite graph. A rainbow matching of G is such a matching in which no two edges have the same color. Let G be a properly colored bipartite graph with bipartition ( X , Y ) and . We show that if   = G k   3   7 max , 4 k X Y  , then G has a rainbow coloring of size at least 3 4 k       .

We study dominating sets whose induced subgraphs have a bounded diameter. Such sets were used in recent papers by Kim et al. and Yu et al. to model virtual backbones in wireless networks where the number of hops required to forward messages within the backbone is minimized. We prove that for any fixed k ≥ 1 it is NP-complete to decide whether a given graph admits a dominating set whose induced ...

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 H be a fixed graph on k vertices. For an edge-coloring c of H , we say that H is rainbow, or totally multicolored if c assigns distinct colors to all edges of H . We show, that it is easy to avoid rainbow induced graphs H . Specifically, we prove that for any graph H (with some notable exceptions), and for any graphs G, G 6= H , there is an edge-coloring of G with k colors which contains no...

A path in an edge-colored graph G is rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer k for which there exists a k-edge-coloring of G such that every pair of distinct vertices of G is connected by a rainbow path. Let f(d) denote the minimum number such that rc(G) ≤ f(d) for each bridgeless graph G with diameter d. In this...