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]

Affirmative domination in graphs.

A function f : V → {−1, 0, 1} is an affirmative dominating function of graph G satisfying the conditions that for every vertex u such that f(u) = 0 is adjacent to at least one vertex v for which f(v) = 1 and P u∈N(v) f(u) ≤ 1 for every v ∈ V . The affirmative domination number γa(G) =max{w(f) : f is affirmative dominating function}. In this paper, we initiate the study of affirmative and strongly affirmative dominating functions. Here, we obtain some properties of these new parameters and also determine exact values of some special classes of graph

Minus total domination in graphs

summary:A three-valued function $f\: V\rightarrow \{-1,0,1\}$ defined on the vertices of a graph $G=(V,E)$ is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. That is, for every $v\in V$, $f(N(v))\ge 1$, where $N(v)$ consists of every vertex adjacent to $v$. The weight of an MTDF is $f(V)=\sum f(v)$, over all vertices $v\in V$. The minus total domination number of a graph $G$, denoted $\gamma _t^{-}(G)$, equals the minimum weight of an MTDF of $G$. In this paper, we discuss some properties of minus total domination on a graph $G$ and obtain a few lower bounds for $\gamma _t^{-}(G)$

The frequency assignment problem

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis examines a wide collection of frequency assignment problems. One of the largest topics in this thesis is that of L(2,1)-labellings of outerplanar graphs. The main result in this topic is the fact that there exists a polynomial time algorithm to determine the minimum L(2,1)-span for an outerplanar graph. This result generalises the analogous result for trees, solves a stated open problem and complements the fact that the problem is NP-complete for planar graphs. We furthermore give best possible bounds on the minimum L(2,1)-span and the cyclic-L(2,1)-span in outerplanar graphs, when the maximum degree is at least eight. We also give polynomial time algorithms for solving the standard constraint matrix problem for several classes of graphs, such as chains of triangles, the wheel and a larger class of graphs containing the wheel. We furthermore introduce the concept of one-close-neighbour problems, which have some practical applications. We prove optimal results for bipartite graphs, odd cycles and complete multipartite graphs. Finally we evaluate different algorithms for the frequency assignment problem, using domination analysis. We compute bounds for the domination number of some heuristics for both the fixed spectrum version of the frequency assignment problem and the minimum span frequency assignment problem. Our results show that the standard greedy algorithm does not perform well, compared to some slightly more advanced algorithms, which is what we would expect. In this thesis we furthermore give some background and motivation for the topics being investigated, as well as mentioning several open problems.EPSR

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

The frequency assignment problem

This thesis examines a wide collection of frequency assignment problems. One of the largest topics in this thesis is that of L(2,1)-labellings of outerplanar graphs. The main result in this topic is the fact that there exists a polynomial time algorithm to determine the minimum L(2,1)-span for an outerplanar graph. This result generalises the analogous result for trees, solves a stated open problem and complements the fact that the problem is NP-complete for planar graphs. We furthermore give best possible bounds on the minimum L(2,1)-span and the cyclic-L(2,1)-span in outerplanar graphs, when the maximum degree is at least eight. We also give polynomial time algorithms for solving the standard constraint matrix problem for several classes of graphs, such as chains of triangles, the wheel and a larger class of graphs containing the wheel. We furthermore introduce the concept of one-close-neighbour problems, which have some practical applications. We prove optimal results for bipartite graphs, odd cycles and complete multipartite graphs. Finally we evaluate different algorithms for the frequency assignment problem, using domination analysis. We compute bounds for the domination number of some heuristics for both the fixed spectrum version of the frequency assignment problem and the minimum span frequency assignment problem. Our results show that the standard greedy algorithm does not perform well, compared to some slightly more advanced algorithms, which is what we would expect. In this thesis we furthermore give some background and motivation for the topics being investigated, as well as mentioning several open problems.EThOS - Electronic Theses Online ServiceEPSRCGBUnited Kingdo