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...

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...

Connected dominating set based routing is a promising approach for enhancing the routing efficiency in the mobile ad hoc networks. However, finding the minimum dominating set in an arbitrary graph is an NP-complete problem. Restricted Rule k is a localized algorithm for finding a small dominating set in mobile ad hoc networks. It starts with a large initial dominating set and uses the local inf...

Perhaps the best known kernelization result is the kernel of size 335k for the Planar Dominating Set problem by Alber et al. [1], later improved to 67k by Chen et al. [5]. This result means roughly, that the problem of finding the smallest dominating set in a planar graph is easy when the optimal solution is small. On the other hand, it is known that Planar Dominating Set parameterized by k = |...

We consider a connected undirected graph G(n,m) with n nodes and m edges. A k-dominating set D in G is a set of nodes having the property that every node in G is at most k edges away from at least one node in D. Finding a k-dominating set of minimum size is NP-hard. We give a new synchronous distributed algorithm to find a k-dominating set in G of size no greater than ⌊n/(k + 1)⌋. Our algorithm...