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>Magister Scientiae - MSc / A Steiner triple system of order v is a collection of subsets of size three from a set of v-elements such that every pair of the elements of the set is contained in exactly one 3-subset. In this study, we discuss some known Steiner triple systems and their automorphism groups. We also construct block intersection graphs of the Steiner triple systems of our consideration and compare their automorphism groups to the automorphism groups of the Steiner triple systems.

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In this thesis we study two clique-cover problems which have interesting applications regarding the k -bend intersection graph representation: the edge-clique-cover-degree problem and the edge-clique-layered-cover problem. We focus on the complexity of these problems and polynomial time algorithms on restricted classes of graphs. The main results of the thesis are NP-completness of the edge-clique-layered-cover problem and a polynomial-time 2-approximation algorithm on the subclass of diamond-free graphs for the same problem as well as some upper bounds on particular graph classes.

Geometrically representable graphs are extensively studied area of research in contempo- rary literature due to their structural characterizations and efficient algorithms. The most frequently studied class of such graphs is the class of interval graphs. In this thesis we focus on two problems, generalizing the problem of recognition, for classes related to interval graphs. In the first part, we are concerned with adjusted interval graphs. This class has been studied as the right digraph analogue of interval graphs. For interval graphs, there are polynomial algorithms to extend a partial representation by given intervals into a full interval representation. We will introduce a similar problem - the partial ordering extension - and we will provide a polynomial algorithm to extend a partial ordering of adjusted interval digraphs. In the second part, we show two NP-completeness results regarding the simultaneous representation problem, introduced by Lubiw and Jampani. The simultaneous representation problem for a given class of intersection graphs asks if some k graphs can be represented so that every vertex is represented by the same object in each representation. We prove that it is NP-complete to decide this for the class of interval and circular-arc graphs in the case when k is a part of the input and graphs...

A graph Gis said to be an intersection graph of sets from a family of sets if there exists a function ƒ : V(G)→ such that for u,v V(G), (u,v) E(G) ƒ (u) ƒ (v) ≠ . Interval graphs are thus the intersection graphs of closed intervals on the real line and unit interval graphs are the intersection graphs of unit length intervals on the real line. An interval on the real line can be generalized to a “kbox” in Rk.A kbox B =(R1,R2,...,Rk), where each Riis a closed interval on the real line, is defined to be the Cartesian product R1x R2x…x Rk. If each Ri is a unit length interval, we call B a k-cube. Thus, 1-boxes are just closed intervals on the real line whereas 2-boxes are axis-parallel rectangles in the plane. We study the intersection graphs of k-boxes and k-cubes. The parameter boxicity of a graph G, denoted as box(G), is the minimum integer k such that G is an intersection graph of k-boxes. Similarly, the cubicity of G, denoted as cub(G), is the minimum integer k such that G is an intersection graph of k-cubes. Thus, interval graphs are the graphs with boxicity at most 1 and unit interval graphs are the graphs with cubicity at most 1. These parameters were introduced by F. S.Roberts in 1969. In some sense, the boxicity of a graph is a measure of how different a graph is from an interval graph and in a similar way, the cubicity is a measure of how different the graph is from a unit interval graph. We prove several upper bounds on the boxicity and cubicity of general as well as special classes of graphs in terms of various graph parameters such as the maximum degree, the number of vertices and the bandwidth. The following are some of the main results presented. 1. We show that for any graph G with maximum degree Δ , box(G)≤ Δ 22 . This result implies that bounded degree graphs have bounded boxicity no matter how large the graph might be. 2. It was shown in [18] that the boxicity of a graph on n vertices with maximum degree Δ is O(Δ ln n). But a similar bound does not hold for the average degree davof a graph. [18] gives graphs in which the boxicity is exponentially larger than davln n. We show that even though an O(davln n) upper bound for boxicity does not hold for all graphs, for almost all graphs, boxicity is O(davln n). 3. The ratio of the cubicity to boxicity of any graph shown in [15] when combined with the results on boxicity show that cub(G) is O(Δ ln 2 n) and O(2 ln n) for any graph G on n vertices and with maximum degree . By using a randomized construction, we prove the better upper bound cub(G) ≤ [4(Δ + 1) ln n.] 4. Two results relating the cubicity of a graph to its bandwidth b are presented. First, it is shown that cub(G) ≤ 12(Δ + 1)[ ln(2b)] + 1. Next, we derive the upper bound cub(G) ≤ b + 1. This bound is used to derive new upper bounds on the cubicity of special graph classes like circular arc graphs, cocomparability graphs and ATfree graphs in relation to the maximum degree. 5. The upper bound for cubicity in terms of the bandwidth gives an upper bound of Δ + 1 for the cubicity of interval graphs. This bound is improved to show that for any interval graph G with maximum degree , cub(G) ≤[ log2 Δ] + 4. 6. Scheinerman [54] proved that the boxicity of any outerplanar graph is at most 2. We present an independent proof for the same theorem. 7. Halin graphs are planar graphs formed by adding a cycle connecting the leaves of a tree none of whose vertices have degree 2. We prove that the boxicity of any Halin graph is equal to 2 unless it is a complete graph on 4 vertices, in which case its boxicity is 1.

Given a set of unit disks in the plane with union area A, what fraction of A can be covered by selecting a pairwise disjoint subset of the disks? Richard Rado conjectured 1/4 and proved 1/4.41. In this thesis, we consider a variant of this problem where the disjointness constraint is relaxed: selected disks must be k-colourable with disks of the same colour pairwise-disjoint. Rado's problem is then the case where k = 1, and we focus our investigations on what can be proven for k > 1. Motivated by the problem of channel-assignment for Wi-Fi wireless access points, in which the use of 3 or fewer channels is a standard practice, we show that for k = 3 we can cover at least 1/2.09 and for k = 2 we can cover at least 1/2.82. We present a randomized algorithm to select and colour a subset of n disks to achieve these bounds in O(n) expected time. To achieve the weaker bounds of 1/2.77 for k = 3 and 1/3.37 for k = 2 we present a deterministic O(n^2) time algorithm. We also look at what bounds can be proven for arbitrary k, presenting two different methods of deriving bounds for any given k and comparing their performance. One of our methods is an extension of the method used to prove bounds for k = 2 and k = 3 above, while the other method takes a novel approach. Rado's proof is constructive, and uses a regular lattice positioned over the given set of disks to guide disk selection. Our proofs are also constructive and extend this idea: we use a k-coloured regular lattice to guide both disk selection and colouring. The complexity of implementing many of the constructions used in our proofs is dominated by a lattice positioning step. As such, we discuss the algorithmic issues involved in positioning lattices as required by each of our proofs. In particular, we show that a required lattice positioning step used in the deterministic O(n^2) algorithm mentioned above is 3SUM-hard, providing evidence that this algorithm is optimal among algorithms employing such a lattice positioning approach. We also present evidence that a similar lattice positioning step used in the constructions for our better bounds for k = 2 and k = 3 may not have an efficient exact implementation.

Given a set of unit disks in the plane with union area A, what fraction of A can be covered by selecting a pairwise disjoint subset of the disks? Richard Rado conjectured 1/4 and proved 1/4.41. In this thesis, we consider a variant of this problem where the disjointness constraint is relaxed: selected disks must be k-colourable with disks of the same colour pairwise-disjoint. Rado's problem is then the case where k = 1, and we focus our investigations on what can be proven for k > 1. Motivated by the problem of channel-assignment for Wi-Fi wireless access points, in which the use of 3 or fewer channels is a standard practice, we show that for k = 3 we can cover at least 1/2.09 and for k = 2 we can cover at least 1/2.82. We present a randomized algorithm to select and colour a subset of n disks to achieve these bounds in O(n) expected time. To achieve the weaker bounds of 1/2.77 for k = 3 and 1/3.37 for k = 2 we present a deterministic O(n^2) time algorithm. We also look at what bounds can be proven for arbitrary k, presenting two different methods of deriving bounds for any given k and comparing their performance. One of our methods is an extension of the method used to prove bounds for k = 2 and k = 3 above, while the other method takes a novel approach. Rado's proof is constructive, and uses a regular lattice positioned over the given set of disks to guide disk selection. Our proofs are also constructive and extend this idea: we use a k-coloured regular lattice to guide both disk selection and colouring. The complexity of implementing many of the constructions used in our proofs is dominated by a lattice positioning step. As such, we discuss the algorithmic issues involved in positioning lattices as required by each of our proofs. In particular, we show that a required lattice positioning step used in the deterministic O(n^2) algorithm mentioned above is 3SUM-hard, providing evidence that this algorithm is optimal among algorithms employing such a lattice positioning approach. We also present evidence that a similar lattice positioning step used in the constructions for our better bounds for k = 2 and k = 3 may not have an efficient exact implementation.

The boxicity of a graph G, denoted as box(G), is the minimum dimension d for which each vertex of G can be mapped to a d-dimensional axis-parallel box in Rd such that two boxes intersect if and only if the corresponding vertices of G are adjacent. An axis-parallel box is a generalized rectangle with sides parallel to the coordinate axes. If additionally, we restrict all sides of the rectangle to be of unit length, the new parameter so obtained is called the cubicity of the graph G, denoted by cub(G). F.S. Roberts had shown that for a graph G with n vertices, box(G) ≤ and cub(G) ≤ . A minimum vertex cover of a graph G is a minimum cardinality subset S of the vertex set of G such that each edge of G has at least one endpoint in S. We show that box(G) ≤ +1 and cub(G)≤ t+ ⌈log2(n −t)⌉−1 where t is the cardinality of a minimum vertex cover. Both these bounds are tight. For a bipartite graph G, we show that box(G) ≤ and this bound is tight. We observe that there exist graphs of very high boxicity but with very low chromatic num-ber. For example, there exist bipartite (2 colorable) graphs with boxicity equal to . Interestingly, if boxicity is very close to , then the chromatic number also has to be very high. In particular, we show that if box(G) = −s, s ≥ 02, then x(G) ≥ where X(G) is the chromatic number of G. We also discuss some known techniques for findingan upper boundon the boxicityof a graph -representing the graph as the intersection of graphs with boxicity 1 (boxicity 1 graphs are known as interval graphs) and covering the complement of the graph by co-interval graphs (a co-interval graph is the complement of an interval graph).

Έστω $V$ ένα σύνολο $n$ κορυφών και έστω ${\cal M}$ ένα πεπερασμένα αριθμήσιμο σύνολο $m$ ετικετών. Ένα γράφημα ετικετών προκύπτει αν αντιστοιχήσουμε σε κάθε κορυφή $v \in V$ ένα υποσύνολο $S_v$ του ${\cal M}$ και στη συνέχεια ενώσουμε όποιες κορυφές έχουν κοινά στοιχεία στα αντίστοιχα σύνολα ετικετών τους. Η παρούσα διδακτορική διατριβή ασχολείται με την εξέταση συνδυαστικών ιδιοτήτων και το σχεδιασμό και ανάλυση αλγορίθμων που σχετίζονται με δυο μοντέλα τυχαίων γραφημάτων που προκύπτουν από την επιλογή των συνόλων $S_v$ με βάση συγκεκριμένες κατανομές. Το πρώτο από αυτά τα μοντέλα ονομάζεται \emph{Μοντέλο Τυχαίων Γραφηματων Τομής Ετικετών} ${\cal G}_{n, m, p}$ (\textlatin{random intersection graphs model}) και κάθε σύνολο ετικετών $S_v$ διαμορφώνεται επιλέγοντας ανεξάρτητα κάθε ετικέτα με πιθανότητα $p$. Το δεύτερο μοντέλο ονομάζεται \emph{Ομοιόμορφο Μοντέλο Τυχαίων Γραφηματων Τομής Ετικετών} ${\cal G}_{n, m, \lambda}$ (\textlatin{uniform random intersection graphs model}) και κάθε σύνολο ετικετών $S_v$ επιλέγεται (ανεξάρτητα για κάθε κορυφή) ισοπίθανα ανάμεσα σε όλα τα υποσύνολα του ${\cal M}$ μεγέθους $\lambda$. Τα μοντέλα αυτά μπορούν να χρησιμοποιηθούν για να μοντελοποιήσουν καταστάσεις που αφορούν θέματα ασφάλειας σε δίκτυα αισθητήρων, αλλά και για την αναπαράσταση των συγκρούσεων (\textlatin{conflicts}) που δημιουργούνται σε περιπτώσεις διαμοιρασμού πόρων. Ακόμα, μπορούν να χρησιμοποιηθούν για τη μοντελοποίηση κοινωνικών γραφημάτων (\textlatin{social graphs}) στα οποία δυο οντότητες συνδέονται όταν έχουν κάποιο κοινό χαρακτηριστικό. Στο Μοντέλο Τυχαίων Γραφηματων Τομής Ετικετών ${\cal G}_{n, m, p}$ μελετάμε καταρχήν το πρόβλημα της ύπαρξης κύκλων \textlatin{Hamilton}. Συγκεκριμένα, αποδεικνύουμε ένα άνω φράγμα για την πιθανότητα επιλογής ετικετών $p$ έτσι ώστε κάθε στιγμιότυπο του ${\cal G}_{n, m, p}$ να περιέχει ένα κύκλο \textlatin{Hamilton} με πιθανότητα που τείνει στο 1 καθώς το $n$ τείνει στο άπειρο. Ακόμα, αναλύουμε δυο πιθανοτικούς αλγορίθμους που, για ορισμένες τιμές των παραμέτρων $m, p$ του μοντέλου, καταφέρνουν να κατασκευάσουν ένα κύκλο \textlatin{Hamilton} με πιθανότητα που τείνει στο 1, δηλαδή σχεδόν πάντα. Επίσης, δείχνουμε ότι σχεδόν κάθε στιγμιότυπο του ${\cal G}_{n, m, p}$ έχει καλή επεκτασιμότητα (\textlatin{expansion}), ακόμα και για $p$ πολύ κοντά στο κατώφλι συνεκτικότητας του μοντέλου. Στη συνέχεια, δίνουμε βέλτιστα άνω φράγματα (που ισχύουν με πιθανότητα που τείνει στο 1 σε ένα ευρύ πεδίο τιμών των παραμέτρων του μοντέλου) για σημαντικές ποσότητες που αφορούν τυχαίους περιπάτους σ ε στιγμιότυπα του ${\cal G}_{n, m, p}$ όπως ο χρόνος μίξης (\textlatin{mixing time}) και ο χρόνος κάλυψης (\textlatin{cover time}). Στο Ομοιόμορφο Μοντέλο Τυχαίων Γραφηματων Τομής Ετικετών ${\cal G}_{n, m, \lambda}$ μελετάμε την ύπαρξη κύκλων \textlatin{Hamilton} σε ένα ορισμένο πεδίο τιμών των παραμέτρων $m, \lambda$ του μοντέλου. Τέλος, υπολογίζουμε με τη βοήθεια της Πιθανοτικής Μεθόδου το κατώφλι ύπαρξης ανεξάρτητων συνόλων κορυφών. / Let $V$ be a set of $i$ vertices and let ${\cal M}$ be a finite set of $m$ labels. An intersection graph is then constructed by assigning to each vertex $v \in V$ a subset $S_v$ of ${\cal M}$ and then connecting every pair of vertices that have common labels in their corresponding label sets. This thesis concerns the study of combinatorial properties, as well as the design and analysis of algorithms on two kinds of random intersection graphs models that arise from different choices of the distribution that we use to construct the sets $S_v$. In the first of these models, called \emph{Random Intersection Graphs Model} ${\cal G}_{n, m, p}$, each set of labels $S_v$ is constructed by choosing independently each label with probability $p$. In the second model, called \emph{Uniform Random Intersection Graphs Model} ${\cal G}_{n, m, \lambda}$, each label set $S_v$ is selected equiprobably (and independently for each vertex $v$) among all subsets of ${\cal M}$ of size $\lambda$. These models can be used to abstract situations that concern the efficient and secure communication in sensor networks, but can also be used to model the conflicts that occur in oblivious resource sharing in distributed settings. Moreover, random intersection graph models can be used to model social graphs, in which two entities are connected when they have a common feature. In the Random Intersection Graphs Model ${\cal G}_{n, m, p}$, we first study the existence and efficient construction of Hamilton cycles. More specifically, we give an upper bound for the probability $p$ that is needed for almost every random instance $G_{n, m, p}$ of the model to have a Hamilton cycle. We also present two polynomial time, randomized algorithms for constructing Hamilton cycles in a wide range of the parameters $m, p$. Moreover, we show that almost every random instance of the ${\cal G}_{n, m, p}$ model is an expander, even for $p$ very close to the connectivity threshold. Finally, we give close to optimal bounds (that hold with probability that goes to 1 for a wide range of the parameters of the model) for important quantities (like the mixing time and the cover time) concerning random walks on random instances of ${\cal G}_{n, m, p}$. In the Uniform Random Intersection Graphs Model ${\cal G}_{n, m, \lambda}$ we study the existence of Hamilton cycles for a ce rtain range of the parameters $m, \lambda$. Finally, by using the probabilistic method we compute the independence number of ${\cal G}_{n, m, \lambda}$.