Aspects of functional variations of domination in graphs.

Thesis (Ph.D.)-University of Natal, Pietermaritzburg, 2003.Let G = (V, E) be a graph. For any real valued function f : V >R and SCV, let f (s) = z ues f(u). The weight of f is defined as f(V). A signed k-subdominating function (signed kSF) of G is defined as a function f : V > {-I, I} such that f(N[v]) > 1 for at least k vertices of G, where N[v] denotes the closed neighborhood of v. The signed k-subdomination number of a graph G, denoted by yks-11(G), is equal to min{f(V) I f is a signed kSF of G}. If instead of the range {-I, I}, we require the range {-I, 0, I}, then we obtain the concept of a minus k-subdominating function. Its associated parameter, called the minus k-subdomination number of G, is denoted by ytks-101(G). In chapter 2 we survey recent results on signed and minus k-subdomination in graphs. In Chapter 3, we compute the signed and minus k-subdomination numbers for certain complete multipartite graphs and their complements, generalizing results due to Holm [30]. In Chapter 4, we give a lower bound on the total signed k-subdomination number in terms of the minimum degree, maximum degree and the order of the graph. A lower bound in terms of the degree sequence is also given. We then compute the total signed k-subdomination number of a cycle, and present a characterization of graphs G with equal total signed k-subdomination and total signed l-subdomination numbers. Finally, we establish a sharp upper bound on the total signed k-subdomination number of a tree in terms of its order n and k where 1 < k < n, and characterize trees attaining these bounds for certain values of k. For this purpose, we first establish the total signed k-subdomination number of simple structures, including paths and spiders. In Chapter 5, we show that the decision problem corresponding to the computation of the total minus domination number of a graph is NP-complete, even when restricted to bipartite graphs or chordal graphs. For a fixed k, we show that the decision problem corresponding to determining whether a graph has a total minus domination function of weight at most k may be NP-complete, even when restricted to bipartite or chordal graphs. Also in Chapter 5, linear time algorithms for computing Ytns-11(T) and Ytns-101(T) for an arbitrary tree T are presented, where n = n(T). In Chapter 6, we present cubic time algorithms to compute Ytks-11(T) and Ytks-101l(T) for a tree T. We show that the decision problem corresponding to the computation of Ytks-11(G) is NP-complete, and that the decision problem corresponding to the computation of Ytks-101 (T) is NP-complete, even for bipartite graphs. In addition, we present cubic time algorithms to computeYks-11(T) and Yks-101(T) for a tree T, solving problems appearing in [25]

Discrepancy and Signed Domination in Graphs and Hypergraphs

For a graph G, a signed domination function of G is a two-colouring of the vertices of G with colours +1 and -1 such that the closed neighbourhood of every vertex contains more +1's than -1's. This concept is closely related to combinatorial discrepancy theory as shown by Fueredi and Mubayi [J. Combin. Theory, Ser. B 76 (1999) 223-239]. The signed domination number of G is the minimum of the sum of colours for all vertices, taken over all signed domination functions of G. In this paper, we present new upper and lower bounds for the signed domination number. These new bounds improve a number of known results.Comment: 12 page