As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduced exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertice...

E cient routing among a set of mobile hosts (also called nodes) is one of the most important functions in ad-hoc wireless networks. Routing based on a connected dominating set is a promising approach, where the searching space for a route is reduced to nodes in the set. A set is dominating if all the nodes in the system are either in the set or neighbors of nodes in the set. In this paper, we p...

Let n > 1 be an integer and let G be a graph of order p. A set 9 of vertices of G is a total n-dominating set of G if every vertex of V(G) is within distance n from some vertex of .P’ other than itself. The minimum cardinal@ among all total n-dominating sets of G is called the total n-domination number and is denoted by “J:(G). A set S of vertices of G is n-independent if the distance (in G) be...

Choose random pointsX1, X2, X3, . . . independently from a uniform distribution in a unit ball in <. Call Xn a dominator iff distance(Xn, Xi) ≤ 1 for all i < n, i.e. the first n points are all contained in the unit ball that is centered at the n’th point Xn. We prove that, with probability one, only finitely many of the points are dominators. For the special casem = 2, we consider the unit disk...

Finding a dominating set of minimum cardinality is an NP-hard graph problem, even when the graph is bipartite. In this paper we are interested in solving the problem on graphs having a large independent set. Given a graph G with an independent set of size z, we show that the problem can be solved in time O∗(2n−z), where n is the number of vertices of G. As a consequence, our algorithm is able t...

A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) \ D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set o...

We present fast distributed algorithms for coloring and (connected) dominating set construction in wireless ad hoc networks. We present our algorithms in the context of Unit Disk Graphs which are known to realistically model wireless networks. Our distributed algorithms take into account the loss of messages due to contention from simultaneous interfering transmissions in the wireless medium. W...