The domination number a(G) of a graph G is the size of a minimum dominating set, i.e., a set of points with the property that every other point is adjacent to a point of the set. In general a(G) can be made to increase or decrease by the removal of points from G. Our main objective is the study of this phenomenon.For example we show that if T is a tree with at least three points then a(T -u) > a(T) if and only if u is in every minimum dominating set of 7'. Removal of a set of lines from a graph G cannot decrease the domination number. We obtain some upper bounds on the size of a minimum set of lines which when removed from G increases the domination number.

It is shown that a connected graph G spans an eulerian graph if and only if G is not spanned by an odd complete bigraph K ( 2 m + 1,2n + 1). A disconnected graph spans an eulerian graph if and only if it is not the union of the trivial graph with a complete graph of odd order. Exact formulas are obtained for the number of lines which must be added to such graphs in order to get eulerian graphs.

A well-known model for network reliability studies consists of an undirected graph with perfectly reliable nodes and equal and independent edge failure probabilities. The measure of reliability is then defined to be the probability that the graph is connected. A well-defined synthesis problem is to find the graph that minimizes the failure probability given the number of nodes n , the number of edges e , and the edge failure rate p . In this work, we consider the possibility of the existence of a fixed graph that is optimal for all possible p. It is simple to verify that such graphs exist when e = n -1, or n . Herein, we show that they also exist when e = n + 1 and n + 2.

This paper presents some results regarding the design of reliable networks. The problem under consideration involves networks which are undirected graphs having equal and independent edge failure probabilities. The index of reliability is the probability that the network fails (becomes disconnected). For "small" edge failure probabilities and given p and q there exists a class of p vertex, q edge graphs with the property that any graph in the class has a smaller probability of disconnection than any graph outside of the class. We solve the problem of synthesizing graphs in this class.

The purpose of this article is to introduce several results concerning the analysis and synthesis of reliable or invulnerable networks. First, the notion of signed reliability domination of systems is described and some applications to reliability analysis are reviewed. Then the analysis problem is considered and a brief summary of the difficulty of calculating various reliability measures is presented. Some relevant concepts in the synthesis of a most reliable network are studied. The article concludes with an introduction to a non-probabilistic approach to evaluate the vulnerability of a network.