The k-tuple twin domination in generalized de Bruijn and Kautz networks

AbstractGiven a digraph (network) G=(V,A), a vertex u in G is said to out-dominate itself and all vertices v such that the arc (u,v)∈A; similarly, u in-dominates both itself and all vertices w such that the arc (w,u)∈A. A set D of vertices of G is a k-tuple twin dominating set if every vertex of G is out-dominated and in-dominated by at least k vertices in D, respectively. The k-tuple twin domination problem is to determine a minimum k-tuple twin dominating set for a digraph. In this paper we investigate the k-tuple twin domination problem in generalized de Bruijn networks GB(n,d) and generalized Kautz GK(n,d) networks when d divides n. We provide construction methods for constructing minimum k-tuple twin dominating sets in these networks. These results generalize previous results given by Araki [T. Araki, The k-tuple twin domination in de Bruijn and Kautz digraphs, Discrete Mathematics 308 (2008) 6406–6413] for de Bruijn and Kautz networks

Properties of locally checkable vertex partitioning problems in digraphs

While, for undirected graphs, locally checkable vertex subset and partitioning problems have been studied extensively, the equivalent directed problems have not received nearly as much attention yet. We take a closer look at the relationship between undirected and directed problems considering hardness. We extend some properties that have already been shown for undirected graphs to directed graphs. Furthermore, we explore some of the trivialities in directed problem definitions that do not appear in undirected ones. And finally, we construct and visualize digraph coverings to achieve a deeper understanding of their structure.Masteroppgave i informatikkINF399KMAMN-IN