The queens graph Qn has the squares of then x n chessboard as its vertices; two squares are adjacent if they are in the same row, column or diagonal. A set D of squares of Qn is a dominating set for Qn if every square of Qn is either in D or adjacent to a square in D. If no two squares of a set I are adjacent then I is an independent set. Let 'J'(Qn) denote the minimum size of a dominating set of Qn and let i(Qn) denote the minimum size of an independent dominating set of Qn. The main purpose of this thesis is to determine new values for'!'( Qn). We begin by discussing the most important known lower bounds for 'J'(Qn) in Chapter 2. In Chapter 3 we state the hitherto known values of 'J'(Qn) and explain how they were determined. We briefly explain how to obtain all non-isomorphic minimum dominating sets for Q8 (listed in Appendix A). It is often useful to study these small dominating sets to look for patterns and possible generalisations. In Chapter 4 we determine new values for')' ( Q69 ) , ')' ( Q77 ), ')' ( Q30 ) and i (Q45 ) by considering asymmetric and symmetric dominating sets for the case n = 4k + 1 and in Chapter 5 we search for dominating sets for the case n = 4k + 3, thus determining the values of 'I' ( Q19) and 'I' (Q31 ). In Chapter 6 we prove the upper bound')' (Qn) :s; 1 8 5n + 0 (1), which is better than known bounds in the literature and in Chapter 7 we consider dominating sets on hexagonal boards. Finally, in Chapter 8 we determine the irredundance number for the hexagonal boards H5 and H7, as well as for Q5 and Q6 / Mathematical Sciences / D.Phil. (Applied Mathematics)

Thesis (MSc)--University of Stellenbosch, 2004. / ENGLISH ABSTRACT: Motivation for the study of protection strategies for graphs is rooted in antiquity and has evolved as a subdiscipline of graph theory since the early 1990s. Using, as a point of departure, the notions of weak Roman domination and secure domination (where protection of a graph is required against a single attack) an initial framework for higher order domination was introduced in 2002 (allowing for the protection of a graph against an arbitrary finite, or even infinite, number of attacks). In this thesis, the theory of higher order domination in graphs is broadened yet further to include the possibility of an arbitrary number of guards being stationed at a vertex. The thesis firstly provides a comprehensive survey of the combinatorial literature on Roman domination, weak Roman domination, secure domination and other higher order domination strategies, with a view to summarise the state of the art in the theory of higher order graph domination as at the start of 2004. Secondly, a generalised framework for higher order domination is introduced in two parts: the first catering for the protection of a graph against a finite number of consecutive attacks, and the second concerning the perpetual security of a graph (protection of the graph against an infinite number of consecutive attacks). Two types of higher order domination are distinguished: smart domination (requiring the existence of a protection strategy for any sequence of consecutive attacks of a pre–specified length, but leaving it up to a strategist to uncover such a guard movement strategy for a particular instance of the attack sequence), and foolproof domination (requiring that any possible guard movement strategy be a successful protection strategy for the graph in question). Properties of these higher order domination parameters are examined—first by investigating the application of known higher order domination results from the literature, and secondly by obtaining new results, thereby hopefully improving current understanding of these domination parameters. Thirdly, the thesis contributes by (i) establishing higher order domination parameter values for some special graph classes not previously considered (such as complete multipartite graphs, wheels, caterpillars and spiders), by (ii) summarising parameter values for special graph classes previously established (such as those for paths, cycles and selected cartesian products), and by (iii) improving higher order domination parameter bounds previously obtained (in the case of the cartesian product of two cycles). Finally, a clear indication of unresolved problems in higher order graph domination is provided in the conclusion to this thesis, together with some suggestions as to possibly desirable future generalisations of the theory. / AFRIKAANSE OPSOMMING: Die motivering vir die studie van verdedigingstrategie¨e vir grafieke het sy ontstaan in die antieke wˆereld en het sedert die vroe¨e 1990s as ’n subdissipline in grafiekteorie begin ontwikkel. Deur gebruik te maak van die idee van swak Romynse dominasie en versterkte dominasie (waar verdediging van ’n grafiek teen ’n enkele aanval vereis word) het ’n aanvangsraamwerk vir ho¨er– orde dominasie (wat ’n grafiek teen ’n veelvuldige, of selfs oneindige aantal, aanvalle verdedig) in 2002 die lig gesien. Die teorie van ho¨er–orde dominasie in grafieke word in hierdie tesis verbreed, deur toe te laat dat ’n arbitrˆere aantal wagte by elke punt van die grafiek gestasioneer mag word. Eerstens voorsien die tesis ’n omvangryke oorsig van die kombinatoriese literatuur oor Romynse dominasie, swak Romynse dominasie, versterkte dominasie en ander ho¨er–orde dominasie strategie ¨e, met die doel om die kundigheid betreffende die teorie van ho¨er–orde dominasie, soos aan die begin van 2004, op te som. Tweedens word ’n veralgemeende raamwerk vir ho¨er–orde dominasie bekendgestel, en wel in twee dele. Die eerste deel maak voorsiening vir die verdediging van ’n grafiek teen ’n eindige aantal opeenvolgende aanvalle, terwyl die tweede deel betrekking het op die oneindige sekuriteit van ’n grafiek (verdediging teen ’n oneindige aantal opeenvolgende aanvalle). Daar word tussen twee tipes h¨oer–orde dominasie onderskei: intelligente dominasie (wat slegs die bestaan van ’n verdedigingstrategie vir enige reeks opeenvolgende aanvalle vereis, maar dit aan ’n strateeg oorlaat om ’n suksesvolle bewegingstrategie vir die verdediging teen ’n spesifieke reeks aanvalle te vind), en onfeilbare dominasie (wat vereis dat enige moontlike bewegingstrategie resulteer in ’n suksesvolle verdedigingstrategie vir die betrokke grafiek). Eienskappe van hierdie ho¨er–orde dominasie parameters word ondersoek, deur eerstens die toepasbaarheid van bekende ho¨er–orde dominasie resultate vanuit die literatuur te assimileer, en tweedens nuwe resultate te bekom, in die hoop om die huidige kundigheid met betrekking tot hierdie dominasie parameters te verbreed. Derdens word ’n bydrae gelewer deur (i) ho¨er–orde dominasie parameterwaardes vas te stel vir sommige spesiale klasse grafieke wat nie voorheen ondersoek is nie (soos volledig veelledige grafieke, wiele, ruspers en spinnekoppe), deur (ii) parameterwaardes wat reeds bepaal is (soos byvoorbeeld di´e vir paaie, siklusse en sommige kartesiese produkte) op te som, en deur (iii) bekende ho¨er–orde dominasie parametergrense te verbeter (in die geval van die kartesiese produk van twee siklusse). Laastens word ’n aanduiding van oop probleme in die teorie van ho¨er–orde dominasie in die slothoofstuk van die tesis voorsien, tesame met voorstelle ten opsigte van moontlik sinvolle veralgemenings van die teorie.

Thesis (PhD (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2004. / Suppose a lottery scheme consists of randomly selecting an unordered winning n-subset from a universal set of m numbers, while a player participates in the scheme by purchasing a playing set of any number of unordered n-subsets from the same universal set prior to a winning draw, and is awarded a prize if ...

Thesis (MSc (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2007. / In this thesis we focus on the lower domination parameters of a graph G, denoted ¼(G), for ¼ 2 {i, ir, °}. For each of these parameters, we are interested in characterizing the structure of graphs that are critical when faced with small changes such as vertex-removal, edge-addition and edge-removal. While criticality with respect to independence and domination have been well documented in the literature, many open questions still remain with regards to irredundance. In this thesis we answer some of these questions. First we describe the relationship between transitivity and criticality. This knowledge we then use to determine under which conditions certain classes of graphs are critical. Each of the chosen classes of graphs will provide specific examples of different types of criticality. We also formulate necessary conditions for graphs to be ir-critical and ir-edge-critical.

For a graph G = (V (G), E (G)), a set S ~ V (G) dominates G if each vertex in V (G) \S is adjacent to a vertex in S. The domination number I (G) (independent domination number i (G)) of G is the minimum cardinality amongst its dominating sets (independent dominating sets). G is k-edge-domination-critical, abbreviated k-1- critical, if the domination number k decreases whenever an edge is added. Further, G is hamiltonian if it has a cycle that passes through each of its vertices. This dissertation assimilates research generated by two conjectures: Conjecture I. Every 3-1-critical graph with minimum degree at least two is hamiltonian. Conjecture 2. If G is k-1-critical, then I ( G) = i ( G). The recent proof of Conjecture I is consolidated and presented accessibly. Conjecture 2 remains open for k = 3 and has been disproved for k :::>: 4. The progress is detailed and proofs of new results are presented. / Mathematical Science / M. Sc. (Mathematics)

The queens graph Qn has the squares of then x n chessboard as its vertices; two squares are adjacent if they are in the same row, column or diagonal. A set D of squares of Qn is a dominating set for Qn if every square of Qn is either in D or adjacent to a square in D. If no two squares of a set I are adjacent then I is an independent set. Let 'J'(Qn) denote the minimum size of a dominating set of Qn and let i(Qn) denote the minimum size of an independent dominating set of Qn. The main purpose of this thesis is to determine new values for'!'( Qn). We begin by discussing the most important known lower bounds for 'J'(Qn) in Chapter 2. In Chapter 3 we state the hitherto known values of 'J'(Qn) and explain how they were determined. We briefly explain how to obtain all non-isomorphic minimum dominating sets for Q8 (listed in Appendix A). It is often useful to study these small dominating sets to look for patterns and possible generalisations. In Chapter 4 we determine new values for')' ( Q69 ) , ')' ( Q77 ), ')' ( Q30 ) and i (Q45 ) by considering asymmetric and symmetric dominating sets for the case n = 4k + 1 and in Chapter 5 we search for dominating sets for the case n = 4k + 3, thus determining the values of 'I' ( Q19) and 'I' (Q31 ). In Chapter 6 we prove the upper bound')' (Qn) :s; 1 8 5n + 0 (1), which is better than known bounds in the literature and in Chapter 7 we consider dominating sets on hexagonal boards. Finally, in Chapter 8 we determine the irredundance number for the hexagonal boards H5 and H7, as well as for Q5 and Q6 / Mathematical Sciences / D.Phil. (Applied Mathematics)

Let G = (V, E) be a graph and let S ⊆ V be a subset of vertices. The set S is a defensive alliance if for all x ∈ S, |N[x] ∩ S| ≥ |N[x] − S|. The concept of defensive alliances was introduced in [KHH04], primarily for the modeling of nations in times of war, where allied nations are in mutual agreement to join forces if any one of them is attacked. For a vertex x in a defensive alliance, the number of neighbors of x inside the alliance, plus the vertex x, is at least the number of neighbors of x outside the alliance. In a graph model, the vertices of a graph represent nations and the edges represent country boundaries. Thus, if the nation corresponding to a vertex x is attacked by its neighbors outside the alliance, the attack can be thwarted by x with the assistance of its neighbors in the alliance. In a different subject matter, [FLG00] applies graph theory to model the world wide web, where vertices represent websites and edges represent links between websites. A web community is a subset of vertices of the web graph, such that every vertex in the community has at least as many neighbors in the set as it has outside. So, a web community C satisfies ∀x ∈ C, |N[x] ∩ C| > |N[x] − C|. These sets are very similar to defensive alliances. They are known as strong defensive alliances in the literature of alliances in graphs. Other areas of application for alliances and related topics include classification, data clustering, ecology, business and social networks. iii Consider the application of modeling nations in times of war introduced in the first paragraph. In a defensive alliance, any attack on a single member of the alliance can be successfully defended. However, as will be demonstrated in Chapter 1, a defensive alliance may not be able to properly defend itself when multiple members are under attack at the same time. The concept of secure sets is introduced in [BDH07] for exactly this purpose. The non-empty set S is a secure set if every subset X ⊆ S, with the assistance of vertices in S, can successfully defend against simultaneous attacks coming from vertices outside of S. The exact definition of simultaneous attacks and how such attacks may be defended will be provided in Chapter 1. In [BDH07], the authors presented an interesting characterization for secure sets which resembles the definition of defensive alliances. A non-empty set S is a secure set if and only if ∀X ⊆ S, |N[X] ∩ S| ≥ |N[X] − S| ([BDH07], Theorem 11). The cardinality of a minimum secure set is the security number of G, denoted s(G). A secure set S is a global secure set if it further satisfies N[S] = V . The cardinality of a minimum global secure set of G is the global security number of G, denoted γs(G). In this work, we present results on secure sets and global secure sets. In particular, we treat the computational complexity of finding the security number of a graph, present algorithms and bounds for the global security numbers of trees, and present the exact values of the global security numbers of paths, cycles and their Cartesian products.

Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: The domination number of a graph is the cardinality of a smallest subset of its vertex set with the property that each vertex of the graph is in the subset or adjacent to a vertex in the subset. This graph parameter has been studied extensively since its introduction during the early 1960s and finds application in the generic setting where the vertices of the graph denote physical entities that are typically geographically dispersed and have to be monitored efficiently, while the graph edges model links between these entities which enable guards, stationed at the vertices, to monitor adjacent entities. In the above application, the guards remain stationary at the entities. In 2005, this constraint was, however, relaxed by the introduction of a new domination-related parameter, called the secure domination number. In this relaxed, dynamic setting, each unoccupied entity is defended by a guard stationed at an adjacent entity who can travel along an edge to the unoccupied entity in order to resolve a security threat that may occur there, after which the resulting configuration of guards at the entities is again required to be a dominating set of the graph. The secure domination number of a graph is the smallest number of guards that can be placed on its vertices so as to satisfy these requirements. In this generalised setting, the notion of edge removal is important, because one might seek the cost, in terms of the additional number of guards required, of protecting the complex of entities modelled by the graph if a number of edges in the graph were to fail (i.e. a number of links were to be eliminated form the complex, thereby disqualifying guards from moving along such disabled links). A comprehensive survey of the literature on secure graph domination is conducted in this dissertation. Descriptions of related, generalised graph protection parameters are also given. The classes of graphs with secure domination number 1, 2 or 3 are characterised and a result on the number of defenders in any minimum secure dominating set of a graph without end-vertices is presented, after which it is shown that the decision problem associated with computing the secure domination number of an arbitrary graph is NP-complete. Two exponential-time algorithms and a binary programming problem formulation are presented for computing the secure domination number of an arbitrary graph, while a linear algorithm is put forward for computing the secure domination number of an arbitrary tree. The practical efficiencies of these algorithms are compared in the context of small graphs. The smallest and largest increase in the secure domination number of a graph are also considered when a fixed number of edges are removed from the graph. Two novel cost functions are introduced for this purpose. General bounds on these two cost functions are established, and exact values of or tighter bounds on the cost functions are determined for various infinite classes of special graphs. Threshold information is finally established in respect of the number of possible edge removals from a graph before increasing its secure domination number. The notions of criticality and stability are introduced and studied in this respect, focussing on the smallest number of arbitrary edges whose deletion necessarily increases the secure domination number of the resulting graph, and the largest number of arbitrary edges whose deletion necessarily does not increase the secure domination number of the resulting graph. / AFRIKAANSE OPSOMMING: Die dominasiegetal van ’n grafiek is die kardinaalgetal van ’n kleinste deelversameling van die grafiek se puntversameling met die eienskap dat elke punt van die grafiek in die deelversameling is of naasliggend is aan ’n punt in die deelversameling. Hierdie grafiekparameter is sedert die vroeë 1960s uitvoerig bestudeer en vind toepassing in die generiese situasie waar die punte van die grafiek fisiese entiteite voorstel wat tipies geografies verspreid is en doeltreffend gemonitor moet word, terwyl die lyne van die grafiek skakels tussen hierdie entiteite voorstel waarlangs wagte, wat by die entiteite gebaseer is, naasliggende entiteite kan monitor. In die bogenoemde toepassing, bly die wagte bewegingloos by die fisiese entiteite waar hulle geplaas word. In 2005 is hierdie beperking egter verslap met die daarstelling van ’n nuwe dominasie-verwante grafiekparameter, bekend as die sekure dominasiegetal. In hierdie verslapte, dinamiese situasie word elke punt sonder ’n wag deur ’n wag verdedig wat by ’n naasliggende punt geplaas is en wat langs die verbindingslyn na die leë punt kan beweeg om daar ’n bedreiging te neutraliseer, waarna die gevolglike plasing van wagte weer ’n dominasieversameling van die grafiek moet vorm. Die sekure dominasiegetal van ’n grafiek is die kleinste getal wagte wat op die punte van die grafiek geplaas kan word om aan hierdie vereistes te voldoen. Die beginsel van lynverwydering speel ’n belangrike rol in hierdie veralgemeende situasie, omdat daar gevra mag word na die koste, in terme van die addisionele getal wagte wat vereis word, om die kompleks van entiteite wat deur die grafiek gemodelleer word, te beveilig indien ’n aantal lynfalings in die grafiek plaasvind (m.a.w. indien ’n aantal skakels uit die kompleks van entiteite verwyder word, en wagte dus nie meer langs sulke skakels mag beweeg nie). ’n Omvattende literatuurstudie oor sekure dominasie van grafieke word in hierdie verhandeling gedoen. Beskrywings van verwante, veralgemeende verdedigingsparameters in grafiekteorie word ook gegee. Die klasse van grafieke met sekure dominasiegetal 1, 2 of 3 word gekarakteriseer en ’n resultaat oor die getal verdedigers in enige kleinste sekure dominasieversameling van ’n grafiek sonder endpunte word daargestel, waarna daar getoon word dat die beslissingsprobleem onderliggend aan die berekening van die sekure dominasiegetal van ’n arbitrêre grafiek NP- volledig is. Twee eksponensiële-tyd algoritmes en ’n binêre programmeringsformulering word vir die bepaling van die sekure dominasiegetal van ’n arbitrêre grafiek daargestel, terwyl ’n lineêre algoritme vir die berekening van die sekure dominasiegetal van ’n arbitrêre boom ontwerp word. Die praktiese doeltreffendhede van hierdie algoritmes word vir klein grafieke met mekaar vergelyk. Die kleinste en groostste toename in die sekure dominasiegetal van ’n grafiek word ook oorweeg wanneer ’n vaste getal lyne uit die grafiek verwyder word. Twee nuwe kostefunksies word vir hierdie doel daargestel en algemene grense word op hierdie kostefunksies vir arbitrêre grafieke bepaal, terwyl eksakte waardes van of verbeterde grense op hierdie kostefunksies vir verskeie oneindige klasse van spesiale grafieke bereken word. Drempelinligting word uiteindelik bepaal in terme van die moontlike getal lynverwyderings uit ’n grafiek voordat die sekure dominasiegetal daarvan toeneem. Die konsepte van kritiekheid en stabiliteit word in hierdie konteks bestudeer, met ’n fokus op die kleinste getal arbitrêre lynfalings wat noodwendig die sekure dominasiegetal van die gevolglike grafiek laat toeneem, of die grootste getal arbitrêre lynfalings wat noodwendig die sekure dominasiegetal van die gevolglike grafiek onveranderd laat.