Charles L. Suffel

Domination alteration sets in graphs

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 - v) > @a (T) if and only if @n is in every minimum dominating set of T. 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.

Combinatorial optimization problems in the analysis and design of probabilistic networks

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 spanning subgraphs of eulerian graphs

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(2m + 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.

The complexity of the residual node connectedness reliability problem

This paper considers a probabilistic network in which the edges are perfectly reliable but the nodes fail with some known probabilities. The network is in an operational state if the surviving nodes induce a connected graph. The residual node connectedness reliability

On the existence of uniformly optimally reliable networks

R(G)

On the characterization of graphs with maximum number of spanning trees

of a network G is the probability that the graph induced by the surviving nodes is connected. This reliability measure is very different from the widely studied K-terminal network reliability measure. It is proven that the problem of computing the residual connectedness reliability is NP-hard by showing that the problem of counting the number of node induced connected subgraphs of a given graph is

Computing residual connectedness reliability for restricted networks

# {\bf P}

A reliability-improving graph transformation with applications to network reliability

-complete. The problem remains

Least reliable networks and the reliability domination

# {\bf P}

Some Alternate Characterizations of Reliability Domination

-complete for split graphs as well as planar and bipartite graphs.