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

Zero forcing in iterated line digraphs

Zero forcing is a propagation process on a graph, or digraph, defined in linear algebra to provide a bound for the minimum rank problem. Independently, zero forcing was introduced in physics, computer science and network science, areas where line digraphs are frequently used as models. Zero forcing is also related to power domination, a propagation process that models the monitoring of electrical power networks. In this paper we study zero forcing in iterated line digraphs and provide a relationship between zero forcing and power domination in line digraphs. In particular, for regular iterated line digraphs we determine the minimum rank/maximum nullity, zero forcing number and power domination number, and provide constructions to attain them. We conclude that regular iterated line digraphs present optimal minimum rank/maximum nullity, zero forcing number and power domination number, and apply our results to determine those parameters on some families of digraphs often used in applications

Private Out-Domination Number of Generalized de Bruijn Digraphs

Dominating sets are widely applied in the design and efficient use of computer networks. They can be used to decide the placement of limited resources, so that every node has access to the resource through neighbouring node. The most efficient solution is one that avoids duplication of access to the resources. This more restricted version of minimum dominating set is called an private dominating set. A vertex v in a digraph D is called a private out-neighbor of the vertex u in S (subset of V(D)) if u is the only element in the intersection of in-neighborhood set of v and S. A subset S of the vertex set V (D) of a digraph D is called a private out-dominating set of D if every vertex of V − S is a private out-neighbor of some vertex of S. The minimum cardinality of a private out-dominating set is called the private out-domination number. In this paper, we investigate the private out-domination number of generalized de Bruijn digraphs. We estabilsh the bounds of private out-domination number. Finally, we present exact values and sharp upperbounds of private out-domination number of some classes of generalized de Bruijn digraphs

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

On the domination number of t-constrained de Bruijn graphs

International audienceMotivated by the work on the domination number of directed de Bruijn graphs and some of its generalizations, in this paper we introduce a natural generalization of de Bruijn graphs (directed and undirected), namely t-constrained de Bruijn graphs, where t is a positive integer, and then study the domination number of these graphs. Within the definition of t-constrained de Bruijn graphs, de Bruijn and Kautz graphs correspond to 1-constrained and 2-constrained de Bruijn graphs, respectively. This generalization inherits many structural properties of de Bruijn graphs and may have similar applications in interconnection networks or bioinformatics. We establish upper and lower bounds for the domination number on t-constrained de Bruijn graphs both in the directed and in the undirected case. These bounds are often very close and in some cases we are able to find the exact value

The watchman's walk problem on directed graphs

A watchman’s walk in a graph is a minimum closed dominating walk. Given a graph and a single watchman, the aim of the Watchman’s walk problem is to find a shortest closed walk that allows the guard to efficiently monitor all vertices in the graph. In a directed graph, a watchman’s walk must obey the direction of the arcs. In this case, we say that the guard can only move to and see the vertices that are adjacent to him relative to outgoing arcs. In this thesis, we consider the watchman’s walk problem on directed graphs. In particular, we study the problem on tournaments, orientations of complete bipartite and multipartite graphs, and directed graphs formed from sequences