Parameters related to fractional domination in graphs.

Thesis (M.Sc.)-University of Natal, 1995.The use of characteristic functions to represent well-known sets in graph theory such as dominating, irredundant, independent, covering and packing sets - leads naturally to fractional versions of these sets and corresponding fractional parameters. Let S be a dominating set of a graph G and f : V(G)~{0,1} the characteristic function of that set. By first translating the restrictions which define a dominating set from a set-based to a function-based form, and then allowing the function f to map the vertex set to the unit closed interval, we obtain the fractional generalisation of the dominating set S. In chapter 1, known domination-related parameters and their fractional generalisations are introduced, relations between them are investigated, and Gallai type results are derived. Particular attention is given to graphs with symmetry and to products of graphs. If instead of replacing the function f : V(G)~{0,1} with a function which maps the vertex set to the unit closed interval we introduce a function f' which maps the vertex set to {0, 1, ... ,k} (where k is some fixed, non-negative integer) and a corresponding change in the restrictions on the dominating set, we obtain a k-dominating function. In chapter 2 corresponding k-parameters are considered and are related to the classical and fractional parameters. The calculations of some well known fractional parameters are expressed as optimization problems involving the k- parameters. An e = 1 function is a function f : V(G)~[0,1] which obeys the restrictions that (i) every non-isolated vertex u is adjacent to some vertex v such that f(u)+f(v) = 1, and every isolated vertex w has f(w) = 1. In chapter 3 a theory of e = 1 functions and parameters is developed. Relationships are traced between e = 1 parameters and those previously introduced, some Gallai type results are derived for the e = 1 parameters, and e = 1 parameters are determined for several classes of graphs. The e = 1 theory is applied to derive new results about classical and fractional domination parameters

On the integrity of domination in graphs.

Thesis (M.Sc.)-University of Natal, 1993.This thesis deals with an investigation of the integrity of domination in a.graph, i.e., the extent to which domination properties of a graph are preserved if the graph is altered by the deletion of vertices or edges or by the insertion of new edges. A brief historical introduction and motivation are provided in Chapter 1. Chapter 2 deals with kedge-( domination-)critical graphs, i.e., graphsG such that )'(G) = k and )'(G+e) < k for all e E E(G). We explore fundamental properties of such graphs and their characterization for small values of k. Particular attention is devoted to 3-edge-critical graphs. In Chapter 3, the changes in domination number brought aboutby vertex removal are investigated. \ Parameters )'+'(G) (and "((G)), denoting the smallest number of vertices of G in a set 5 such that )'(G-5) > )'(G) ()'(G -5) < )'(G), respectively), are investigated, as are'k-vertex-critical graphs G (with )'(G) = k and )'(G-v) < k for all v E V(O)). The existence of smallest'domination-forcing sets of vertices of graphs is considered. The bondage number 'Y+'(G), i.e., the smallest number of edges of a graph G in a set F such that )'(G- F) > )'(0), is investigated in Chapter 4, as are associated extremal graphs. Graphs with dominating sets or domination numbers that are insensitive to the removal of an arbitrary edge are considered, with particular reference to such graphs of minimum size. Finally, in Chapter 5, we-discuss n-dominating setsD of a graph G (such that each vertex in G-D is adjacent to at least n vertices in D) and associated parameters. All chapters but the first and fourth contain a listing of unsolved problems and conjectures