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]

On edge domination numbers of graphs

AbstractLet γs′(G) and γss′(G) be the signed edge domination number and signed star domination number of G, respectively. We prove that 2n-4⩾γss′(G)⩾γs′(G)⩾n-m holds for all graphs G without isolated vertices, where n=|V(G)|⩾4 and m=|E(G)|, and pose some problems and conjectures

The Signed Domination Number of Cartesian Products of Directed Cycles

Let&nbsp;D&nbsp;be a finite simple directed graph with vertex set&nbsp;V(D) and arc set&nbsp;A(D). A function is called a signed dominating function (SDF) iffor each vertex v. The weight&nbsp;w(f&nbsp;) of&nbsp;f&nbsp;is defined by. The signed domination number of a digraph D is&nbsp;gs(D)&nbsp;= min{w(f&nbsp;) :&nbsp;f&nbsp;is an SDF of D}. Let&nbsp;Cmn&nbsp;denotes the Cartesian product of directed cycles of length m and n. In this paper, we determine the exact value of signed domination number of some classes of Cartesian product of directed cycles. In particular, we prove that: (a) gs(C3n) = n if n 0(mod 3), otherwise gs(C3n) = n + 2. (b) gs(C4n) = 2n. (c) gs(C5n) = 2n if n 0(mod 10), gs(C5n) = 2n + 1 if n 3, 5, 7(mod 10), gs(C5n) = 2n + 2 if n 2, 4, 6, 8(mod 10), gs(C5n) = 2n + 3 if n 1,9(mod 10). (d) gs(C6n) = 2n if n 0(mod 3), otherwise gs(C6n) = 2n + 4. (e) gs(C7n) = 3n

Aspects of signed and minus domination in graphs

Ph.D.In Chapter 1 we will give a brief historical account of domination theory and define the necessary concepts which we use in the remainder of the thesis. In Chapter 2 we establish a lower bound for the minus k-subdomination number of trees and characterize those trees which achieve this lower bound. We also compute the value of Yks-101 for comets and for cycles. We then show that the decision problem corresponding to the computation of Yks-101 is NP-complete, even for bipartite graphs. In Chapter 3 we characterize those trees T which achieve the lower bound of Cockayne and Mynhardt, thus generalizing the results of [11] and [2]. We also compute Yks-11 for comets and cycles. In Chapter 4 we study the partial signed domination number of a graph. In particular, we establish a lower bound on Yc/d for regular graphs and prove that the decision problem corresponding to the computation of the partial signed domination number is NP-complete. Chapter 5 features the minus bondage number b- (G) of a nonempty graph G, which is defined as the minimum cardinality of a set of edges whose removal increases the minus domination number of G. We show that the minus bondage and ordinary bondage numbers of a graph are incomparable. Exact values for certain well known classes of graphs are computed and an upper bound for b- is given for trees. Finally, we show that the decision problem corresponding to the computation of b- is N P - hard, even for bipartite graphs. We conclude, in Chapter 6, by discussing possible directions for future research

Aspects of signed and minus domination in graphs

Ph.D.In Chapter 1 we will give a brief historical account of domination theory and define the necessary concepts which we use in the remainder of the thesis. In Chapter 2 we establish a lower bound for the minus k-subdomination number of trees and characterize those trees which achieve this lower bound. We also compute the value of Yks-101 for comets and for cycles. We then show that the decision problem corresponding to the computation of Yks-101 is NP-complete, even for bipartite graphs. In Chapter 3 we characterize those trees T which achieve the lower bound of Cockayne and Mynhardt, thus generalizing the results of [11] and [2]. We also compute Yks-11 for comets and cycles. In Chapter 4 we study the partial signed domination number of a graph. In particular, we establish a lower bound on Yc/d for regular graphs and prove that the decision problem corresponding to the computation of the partial signed domination number is NP-complete. Chapter 5 features the minus bondage number b- (G) of a nonempty graph G, which is defined as the minimum cardinality of a set of edges whose removal increases the minus domination number of G. We show that the minus bondage and ordinary bondage numbers of a graph are incomparable. Exact values for certain well known classes of graphs are computed and an upper bound for b- is given for trees. Finally, we show that the decision problem corresponding to the computation of b- is N P - hard, even for bipartite graphs. We conclude, in Chapter 6, by discussing possible directions for future research