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.

The main results of this dissertation show evidence supporting the Successive Surface Scaffolding Conjecture. This is a new conjecture that, if true, guarantees the existence of all the wye-delta-order minimal obstructions of a surface S as subgraphs of the irreducible triangulations of the surface S with a crosscap added. A new data structure, i.e. an augmented rotation system, is presented and used to create an exponential-time algorithm for embedding graphs in any surface with a constant-time check of the change in genus when inserting an edge. A depiction is a new formal definition for representing an embedding graphically, and it is shown that more than one depiction can be given for nonplanar embeddings, and that sometimes two depictions for the same embedding can be drastically different from each other. An algorithm for finding the essential cycles of an embedding is given, and is used to confirm for the projective-plane obstructions, a theorem that shows any embedding of an obstruction must have every edge in an essential cycle. Obstructions of a general surface S that are minor-minimal and not double-wye-delta-minimal are shown to each have an embedding on the surface S with a crosscap added. Finally, open questions for further research are presented. / Graduate

In the 1970's computer scientists developed the theory of computational complexity. Some problems seemed hard-to-compute, while others were easy. It turned out that many of the hard problems were equally hard in a way that could be precisely specified. They became known as the NP-complete problems. The SATISFIABILITY problem (SAT) was the first problem to be proved NP-complete in 1971. Since then numerous other hard-to-solve problems have been proved to be in NP-complete. In this paper we will examine the problem of how to find a maximum independent set of vertices for a graph. This problem is known as Maximum Independent Set (MIS) for a graph. The corresponding decision problem for MIS is the question, given an integer K, is there a independent set with at least K vertices? This decision problem is INDEPENDENT SET (IS). The intention of this paper is to show through polynomial transformation that IS is in the set of NP-complete Problems. We intend to show that 3SAT is NP-complete and then using this fact, that IS is NP-complete.

In this dissertation, we are interested in studying several parameters of graphs and understanding their extreme values. We begin in Chapter~$2$ with a question on edge colouring. When can a partial proper edge colouring of a graph of maximum degree $\Delta$ be extended to a proper colouring of the entire graph using an `optimal' set of colours? Albertson and Moore conjectured this is always possible provided no two precoloured edges are within distance $2$. The main result of Chapter~$2$ comes close to proving this conjecture. Moreover, in Chapter~$3$, we completely answer the previous question for the class of planar graphs. Next, in Chapter~$4$, we investigate some Ramsey theoretical problems. We determine exactly what minimum degree a graph $G$ must have to guarantee that, for any two-colouring of $E(G)$, we can partition $V(G)$ into two parts where each part induces a connected monochromatic subgraph. This completely resolves a conjecture of Bal and Debiasio. We also prove a `covering' version of this result. Finally, we study another variant of these problems which deals with coverings of a graph by monochromatic components of distinct colours. The following saturation problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger is considered in Chapter~$5$. Given a graph $H$ and a set of colours $\{1,2,\ldots,t\}$ (for some integer $t\geq |E(H)|$), we define $sat_{t}(n, R(H))$ to be the minimum number of $t$-coloured edges in a graph on $n$ vertices which does not contain a rainbow copy of $H$ but the addition of any non-edge in any colour from $\{1,2,\ldots,t\}$ creates such a copy. We prove several results concerning these extremal numbers. In particular, we determine the correct order of $sat_{t}(n, R(H))$, as a function of $n$, for every connected graph $H$ of minimum degree greater than $1$ and for every integer $t\geq e(H)$. In Chapter~$6$, we consider the following question: under what conditions does a Hamiltonian graph on $n$ vertices possess a second cycle of length at least $n-o(n)$? We prove that the `weak' assumption of a minimum degree greater or equal to $3$ guarantees the existence of such a long cycle. We solve two problems related to majority colouring in Chapter~$7$. This topic was recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised the problem of determining, for a natural number $k$, the smallest positive integer $m = m(k)$ such that every digraph can be coloured with $m$ colours, where each vertex has the same colour as at most a proportion of $\frac{1}{k}$ of its out-neighbours. Our main theorem states that $m(k) \in \{2k-1, 2k\}$. We study the following problem, raised by Caro and Yuster, in Chapter~$8$. Does every graph $G$ contain a `large' induced subgraph $H$ which has $k$ vertices of degree exactly $\Delta(H)$? We answer in the affirmative an approximate version of this question. Indeed, we prove that, for every $k$, there exists $g(k)$ such that any $n$ vertex graph $G$ with maximum degree $\Delta$ contains an induced subgraph $H$ with at least $n-g(k)\sqrt{\Delta}$ vertices such that $V(H)$ contains at least $k$ vertices of the same degree $d \ge \Delta(H)-g(k)$. This result is sharp up to the order of $g(k)$. %Subsequently, we investigate a concept called $\textit{path-pairability}$. A graph is said to be path-pairable if for any pairing of its vertices there exist a collection of edge-disjoint paths routing the the vertices of each pair. A question we are concerned here asks whether every planar path pairable graph on $n$ vertices must possess a vertex of degree linear in $n$. Indeed, we answer this question in the affirmative. We also sketch a proof resolving an analogous question for graphs embeddable on surfaces of bounded genus. Finally, in Chapter~$9$, we move on to examine $k$-linked tournaments. A tournament $T$ is said to be $k$-linked if for any two disjoint sets of vertices $\{x_1,\ldots ,x_k\}$ and $\{y_1,\dots,y_k\}$ there are directed vertex disjoint paths $P_1,\dots, P_k$ such that $P_i$ joins $x_i$ to $y_i$ for $i = 1,\ldots, k$. We prove that any $4k$ strongly-connected tournament with sufficiently large minimum out-degree is $k$-linked. This result comes close to proving a conjecture of Pokrovskiy.

In a 2011 paper by Beeler and Hoilman, the game of peg solitaire is generalized to arbitrary boards. These boards are treated as graphs in the combinatorial sense. An open problem from that paper is to determine the minimum number of edges necessary for a graph with a fixed number of vertices to be solvable. This thesis provides new bounds on this number. It also provides necessary and sufficient conditions for two families of graphs to be solvable, along with criticality results, and the maximum number of pegs that can be left in each of the two graph families.

A 2004 article written by Yixun Lin and Aifeng Yang published in the journal Discrete Math characterized the set of a 3-cutwidth critical graphs by five specified elements. This project extends the idea to 4-cutwidth critical graphs.

A 3-map is a 3-region colorable map. They have been studied by Craft and White in their paper 3-maps. This thesis introduces topological graph theory and then investigates 3-maps in detail, including examples, special types of 3-maps, the use of 3-maps to find the genus of special graphs, and a generalization known as n-maps.

The property that a graph has an embedding in the projective plane is closed under taking minors. Thus by the well known Graph Minor theorem of Robertson and Seymour, there exists a finite list of minor-minimal graphs, call it L, such that a given graph G is projective planar if and only if G does not contain any graph isomorphic to a member of L as a minor. Glover, Huneke and Wang found 35 graphs in L, and Archdeacon proved that those are all the members of L, but Archdeacon's proof never appeared in any refereed journal. In this thesis we develop a modern approach and technique for finding the list L, independent of previous work. Our approach is based on conditioning on the connectivity of a member of L. Assume G is a member of L. If G is not 3-connected then the structure of G is well understood. In the case that G is 3-connected, the problem breaks down into two main cases, either G has an internal separation of order three or G is internally 4-connected. In this thesis we find the set of all 3-connected minor minimal non-projective planar graphs with an internal 3-separation. For proving our main result, we use a technique which can be considered as a variation and generalization of the method that Robertson, Seymour and Thomas used for non-planar extension of planar graphs. Using this technique, besides our main result, we also classify the set of minor minimal obstructions for a-, ac-, abc-planarity for rooted graphs. (A rooted graph (G,a,b,c) is a-planar if there exists a split of the vertex a to a' and a' in G such that the new graph G' obtained by the split has an embedding in a disk such that the vertices a', b, a', c are on the boundary of the disk in the order listed. We define b- and c-planarity analogously. We say that the rooted graph (G,a,b,c) is ab-planar if it is a-planar or b-planar, and we define abc-planarity analogously.)

The first result of this thesis is a generation theorem for bricks. A brick is a 3-connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of a decomposition procedure of Kotzig, and Lovasz and Plummer. We prove that every brick except for the Petersen graph can be generated from K_4 or the prism by repeatedly applying certain operations in such a way that all the intermediate graphs are bricks. We use this theorem to prove an exact upper bound on the number of edges in a minimal brick with given number of vertices and to prove that every minimal brick has at least three vertices of degree three. The second half of the thesis is devoted to an investigation of graphs that admit Pfaffian orientations. We prove that a graph admits a Pfaffian orientation if and only if it can be drawn in the plane in such a way that every perfect matching crosses itself even number of times. Using similar techniques, we give a new proof of a theorem of Kleitman on the parity of crossings and develop a new approach to Turan's problem of estimating crossing number of complete bipartite graphs. We further extend our methods to study k-Pfaffian graphs and generalize a theorem by Gallucio, Loebl and Tessler. Finally, we relate Pfaffian orientations and signs of edge-colorings and prove a conjecture of Goddyn that every k-edge-colorable k-regular Pfaffian graph is k-list-edge-colorable. This generalizes a theorem of Ellingham and Goddyn for planar graphs.

Our world is filled with large-scale engineering systems, which provide various services and conveniences in our daily life. A distinctive trend in the development of today's large-scale engineering systems is the extensive and aggressive adoption of automation and autonomy that enable the significant improvement of systems' robustness, efficiency, and performance, with considerably reduced manning and maintenance costs, and the U.S. Navy's DD(X), the next-generation destroyer program, is considered as an extreme example of such a trend. This thesis pursues a modeling solution for performing simulation-based analysis in the conceptual or preliminary design stage of an intelligent, self-reconfigurable ship fluid system, which is one of the concepts of DD(X) engineering plant development. Through the investigations on the Navy's approach for designing a more survivable ship system, it is found that the current naval simulation-based analysis environment is limited by the capability gaps in damage modeling, dynamic model reconfiguration, and simulation speed of the domain specific models, especially fluid network models. As enablers of filling these gaps, two essential elements were identified in the formulation of the modeling method. The first one is the graph-based topological modeling method, which will be employed for rapid model reconstruction and damage modeling, and the second one is the recurrent neural network-based, component-level surrogate modeling method, which will be used to improve the affordability and efficiency of the modeling and simulation (M&S) computations. The integration of the two methods can deliver computationally efficient, flexible, and automation-friendly M&S which will create an environment for more rigorous damage analysis and exploration of design alternatives. As a demonstration for evaluating the developed method, a simulation model of a notional ship fluid system was created, and a damage analysis was performed. Next, the models representing different design configurations of the fluid system were created, and damage analyses were performed with them in order to find an optimal design configuration for system survivability. Finally, the benefits and drawbacks of the developed method were discussed based on the result of the demonstration.