Search

A hop dominating set of a graph G is a set D of vertices of G if for every vertex of V (G) \ D, there exists u ∈ D such that d(u, v) = 2. The hop domination number of a graph G, denoted by γh(G), is the minimum cardinality of a hop dominating set of G. We prove that for every tree T of order n with l leaves and s support vertices we have (n − l − s + 4)/3 ≤ γh(G) ≤ n/2, and characterize the trees attaining each of the bounds. [ABSTRACT FROM AUTHOR]

Multi-coloring on the path is a model for frequency assignment in linear cellular networks. Two models have been studied in previous papers: calls may either have finite or infinite duration. For hexagonal networks, a variation of the models where limited frequency reassignment is allowed has also been studied. We add the concept of frequency reassignment to the models of linear cellular networks and close these problems by giving matching upper and lower bounds in all cases. We prove that no randomized algorithm can have a better competitive ratio than the best deterministic algorithms. In addition, we give an exact characterization of the natural greedy algorithms for these problems. All of the above results are with regard to competitive analysis. Taking steps towards a more fine-grained analysis, we consider the case of finite calls and no frequency reassignment and prove that, even though randomization cannot bring the competitive ratio down to one, it seems that the greedy algorithm is expected optimal on uniform random request sequences. We prove this for small paths and indicate it experimentally for larger graphs. [ABSTRACT FROM AUTHOR]