A set S ⊆ V is a dominating set of a graph G = (V, E) if every vertex in V − S is adjacent to at least one vertex in S. The domination number γ(G) of G equals the minimum cardinality of a dominating set S in G; we say that such a set S is a γ-set. In this paper we consider the family of all γ-sets in a graph G and we define the γgraph G(γ) = (V (γ), E(γ)) of G to be the graph whose vertices V (γ) correspond 1-to-1 with the γ-sets of G, and two γ-sets, say D 1 and D 2 , are adjacent in E(γ) if there exists a vertex v ∈ D 1 and a vertex w ∈ D 2 such that v is adjacent to w and D 1 = D 2 − {w} ∪ {v}, or equivalently, D 2 = D 1 − {v} ∪ {w}. In this paper we initiate the study of γ-graphs of graphs.

No abstract

This paper considers the problem of determining minimum spanning trees in networks in which each edge weight can assume a finite number of distinct values. We use the algebraic structure of an underlying Hasse diagram to describe the relationship between different edge-weight realizations of the network, yielding new results on how MSTs change under multiple edge-weight perturbations. We investigate various implementation strategies for updating MSTs in this manner. Computational results are provided for some challenging test networks.

A cylinder graph is the graph Cartesian product of a path and a cycle. In this paper we investigate the length of a minimal spanning tree of a cylinder graph whose edges are assigned random lengths according to independent and uniformly distributed random variables. Our work was inspired by a formula of J. Michael Steele which shows that the expected length of a minimal spanning tree of a connected graph can be calculated through the Tutte polynomial of the graph. First, using transfer matrices, we show how to calculate the Tutte polynomials of cylinder graphs. Second, using Steele's formula, we tabulate the expected lengths of the minimal spanning trees for some cylinder graphs. Third, for a fixed cycle length, we show that the ratio of the expected length of a minimal spanning tree of a cylinder graph to the length of the cylinder graph converges to a constant; this constant is described in terms of the Perron-Frobenius eigenvalue of the accompanying transfer matrix. Finally, we show that the length of a minimal spanning tree of a cylinder graph satisfies a strong law of large numbers.

This paper considers the problem of determining the distribution of the weight W of a minimum spanning tree for an undirected graph with edge weights that are independently distributed discrete random variables. Using the underlying fundamental cutsets and cycles associated with a spanning tree, we are able to obtain upper and lower bounds on the distribution of W. In turn, these are used to establish bounds on E[W]. Our general method for deriving these bounding distributions subsumes existing approximation methods in the literature. Computational results indicate that the new approximation methods provide excellent bounds for some challenging test networks.