Sandra Mitchell 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.

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

Approximating MAPs for belief networks is NP-hard and other theorems

\gamma_P(G)

On the Algorithmic Complexity of Total Domination

. 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

Offensive alliances in graphs

\gamma_P(T)

Self-stabilizing Algorithms for Minimal Dominating Sets and Maximal Independent Sets

in trees T.

Broadcasts in graphs

Abstract Finding maximum a posteriori (MAP) assignments, also called Most Probable Explanations, is an important problem on Bayesian belief networks. Shimony has shown that finding MAPs is NP-hard. In this paper, we show that approximating MAPs with a constant ratio bound is also NP-hard. In addition, we examine the complexity of two related problems which have been mentioned in the literature. We show that given the MAP for a belief network and evidence set, or the family of MAPs if the optimal is not unique, it remains NP-hard to find, or approximate, alternative next-best explanations. Furthermore, we show that given the MAP, or MAPs, for a belief network and an initial evidence set, it is also NP-hard to find, or approximate, the MAP assignment for the same belief network with a modified evidence set that differs from the initial set by the addition or removal of even a single node assignment. Finally, we show that our main result applies to networks with constrained in-degree and out-degree, applies to randomized approximation, and even still applies if the ratio bound, instead of being constant, is allowed to be a polynomial function of various aspects of the network topology.

Domination Subdivision Numbers

A set of vertices D is a dominating set for a graph

Acyclic domination

G = (V,E)