Stephen T. Hedetniemi

A survey of gossiping and broadcasting in communication networks

Gossiping and broadcasting are two problems of information dissemination described for a group of individuals connected by a communication network. In gossiping every person in the network knows a unique item of information and needs to communicate it to everyone else. In broadcasting one individual has an item of information which needs to be communicated to everyone else. We review the results that have been obtained on these and related problems.

Total domination in graphs

A set D of vertices of a finite, undirected graph G = (V, E) is a total dominating set if every vertex of V is adjacent to some vertex of D. In this paper we initiate the study of total dominating sets in graphs and, in particular, obtain results concerning the total domination number of G (the smallest number of vertices in a total dominating set) and the total domatic number of G (the largest order of a partition of G into total dominating sets).

Towards a theory of domination in graphs

This paper presents a quick review of results and applications concerning dominating sets in graphs. The domatic number of a graph is defined and studied. It is seen that the theory of domination resembles the well known theory of colorings of graphs.

Information Dissemination in Trees

In large organizations there is frequently a need to pass information from one place, e.g., the president’s office or company headquarters, to all other divisions, departments or employees. This is often done along organizational reporting lines. Insofar as most organizations are structured in a hierarchical or treelike fashion, this can be described as a process of information dissemination in trees. In this paper we present an algorithm which determines the amount of time required to pass, or to broadcast, a unit of information from an arbitrary vertex to every other vertex in a tree. As a byproduct of this algorithm we determine the broadcast center of a tree, i.e., the set of all vertices from which broadcasting can be accomplished in the least amount of time. It is shown that the subtree induced by the broadcast center of a tree is always a star with two or more vertices. We also show that the problem of determining the minimum amount of time required to broadcast from an arbitrary vertex in an arbit...

Roman domination in graphs

Abstract A Roman dominating function on a graph G=(V,E) is a function f : V→{0,1,2} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. The weight of a Roman dominating function is the value f(V)=∑u∈Vf(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this paper, we study the graph theoretic properties of this variant of the domination number of a graph.

Domination in Graphs Applied to Electric Power Networks

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known vertex covering and dominating set problems in graphs. We consider the graph theoretical representation of this problem as a variation of the dominating set problem and define a set S to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The minimum cardinality of a power dominating set of a graph G is the power domination number

Graphs with forbidden subgraphs

\gamma_P(G)

Minimum broadcast graphs

. We show that the power dominating set (PDS) problem is NP-complete even when restricted to bipartite graphs or chordal graphs. On the other hand, we give a linear algorithm to solve the PDS for trees. In addition, we investigate theoretical properties of

Bibliography on domination in graphs and some basic definitions of domination parameters

\gamma_P(T)

On the Algorithmic Complexity of Total Domination

in trees T.