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]

The Power of Small Coalitions under Two-Tier Majority on Regular Graphs

In this paper, we study the following problem. Consider a setting where a proposal is offered to the vertices of a given network $G$, and the vertices must conduct a vote and decide whether to accept the proposal or reject it. Each vertex $v$ has its own valuation of the proposal; we say that $v$ is ``happy'' if its valuation is positive (i.e., it expects to gain from adopting the proposal) and ``sad'' if its valuation is negative. However, vertices do not base their vote merely on their own valuation. Rather, a vertex $v$ is a \emph{proponent} of the proposal if the majority of its neighbors are happy with it and an \emph{opponent} in the opposite case. At the end of the vote, the network collectively accepts the proposal whenever the majority of its vertices are proponents. We study this problem for regular graphs with loops. Specifically, we consider the class $\mathcal{G}_{n|d|h}$ of $d$-regular graphs of odd order $n$ with all $n$ loops and $h$ happy vertices. We are interested in establishing necessary and sufficient conditions for the class $\mathcal{G}_{n|d|h}$ to contain a labeled graph accepting the proposal, as well as conditions to contain a graph rejecting the proposal. We also discuss connections to the existing literature, including that on majority domination, and investigate the properties of the obtained conditions.Comment: 28 pages, 8 figures, accepted for publication in Discrete Applied Mathematic

Remarks on minus (signed) total domination in graphs

Author name used in this publication: T.C.E. ChengAuthor name also used in this publication: E.F. Shan2007-2008 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe

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

Connected Domination Critical Graphs

This thesis investigates the structure of connected domination critical graphs. The characterizations developed provide an important theoretical framework for addressing a number of difficult computational problems in the areas of operations research (for example, facility locations, industrial production systems), security, communication and wireless networks, transportation and logistics networks, land surveying and computational biology. In these application areas, the problems of interest are modelled by networks and graph parameters such as domination numbers reflect the efficiency and performance of the systems