An upper bound for the crossing number of augmented cubes

A {\it good drawing} of a graph $G$ is a drawing where the edges are non-self-intersecting and each two edges have at most one point in common, which is either a common end vertex or a crossing. The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a good drawing of $G$ in the plane. The {\it $n$-dimensional augmented cube} $AQ_n$, proposed by S.A. Choudum and V. Sunitha, is an important interconnection network with good topological properties and applications. In this paper, we obtain an upper bound on the crossing number of $AQ_n$ less than $26/324^{n}-(2n^2+7/2n-6)2^{n-2}$.Comment: 39 page

Facets of Planar Graph Drawing

This thesis makes a contribution to the field of Graph Drawing, with a focus on the planarity drawing convention. The following three problems are considered. (1) Ordered Level Planarity: We introduce and study the problem Ordered Level Planarity which asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a y-monotone curve. This can be interpreted as a variant of Level Planarity in which the vertices on each level appear in a prescribed total order. We establish a complexity dichotomy with respect to both the maximum degree and the level-width, that is, the maximum number of vertices that share a level. Our study of Ordered Level Planarity is motivated by connections to several other graph drawing problems. With reductions from Ordered Level Planarity, we show NP-hardness of multiple problems whose complexity was previously open, and strengthen several previous hardness results. In particular, our reduction to Clustered Level Planarity generates instances with only two nontrivial clusters. This answers a question posed by Angelini, Da Lozzo, Di Battista, Frati, and Roselli [2015]. We settle the complexity of the Bi-Monotonicity problem, which was proposed by Fulek, Pelsmajer, Schaefer, and Stefankovic [2013]. We also present a reduction to Manhattan Geodesic Planarity, showing that a previously [2009] claimed polynomial time algorithm is incorrect unless P=NP. (2) Two-page book embeddings of triconnected planar graphs: We show that every triconnected planar graph of maximum degree five is a subgraph of a Hamiltonian planar graph or, equivalently, it admits a two-page book embedding. In fact, our result is more general: we only require vertices of separating 3-cycles to have degree at most five, all other vertices may have arbitrary degree. This degree bound is tight: we describe a family of triconnected planar graphs that cannot be realized on two pages and where every vertex of a separating 3-cycle has degree at most six. Our results strengthen earlier work by Heath [1995] and by Bauernöppel [1987] and, independently, Bekos, Gronemann, and Raftopoulou [2016], who showed that planar graphs of maximum degree three and four, respectively, can always be realized on two pages. The proof is constructive and yields a quadratic time algorithm to realize the given graph on two pages. (3) Convexity-increasing morphs: We study the problem of convexifying drawings of planar graphs. Given any planar straight-line drawing of an internally 3-connected graph, we show how to morph the drawing to one with strictly convex faces while maintaining planarity at all times. Our morph is convexity-increasing, meaning that once an angle is convex, it remains convex. We give an efficient algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines. Moreover, we show that a linear number of steps is worst-case optimal.Diese Arbeit behandelt drei unterschiedliche Problemstellungen aus der Disziplin des Graphenzeichnens (Graph Drawing). Bei jedem der behandelten Probleme ist die gesuchte Darstellung planar. (1) Ordered Level Planarity: Wir führen das Problem Ordered Level Planarity ein, bei dem es darum geht, einen Graph so zu zeichnen, dass jeder Knoten an einer vorgegebenen Position der Ebene platziert wird und die Kanten als y-monotone Kurven dargestellt werden. Dies kann als eine Variante von Level Planarity interpretiert werden, bei der die Knoten jedes Levels in einer vorgeschriebenen Reihenfolge platziert werden müssen. Wir klassifizieren die Eingaben bezüglich ihrer Komplexität in Abhängigkeit von sowohl dem Maximalgrad, als auch der maximalen Anzahl von Knoten, die demselben Level zugeordnet sind. Wir motivieren die Ergebnisse, indem wir Verbindungen zu einigen anderen Graph Drawing Problemen herleiten: Mittels Reduktionen von Ordered Level Planarity zeigen wir die NP-Schwere einiger Probleme, deren Komplexität bislang offen war. Insbesondere wird gezeigt, dass Clustered Level Planarity bereits für Instanzen mit zwei nichttrivialen Clustern NP-schwer ist, was eine Frage von Angelini, Da Lozzo, Di Battista, Frati und Roselli [2015] beantwortet. Wir zeigen die NP-Schwere des Bi-Monotonicity Problems und beantworten damit eine Frage von Fulek, Pelsmajer, Schaefer und Stefankovic [2013]. Außerdem wird eine Reduktion zu Manhattan Geodesic Planarity angegeben. Dies zeigt, dass ein bestehender [2009] Polynomialzeitalgorithmus für dieses Problem inkorrekt ist, es sei denn, dass P=NP ist. (2) Bucheinbettungen von dreifach zusammenhängenden planaren Graphen mit zwei Seiten: Wir zeigen, dass jeder dreifach zusammenhängende planare Graph mit Maximalgrad 5 Teilgraph eines Hamiltonischen planaren Graphen ist. Dies ist äquivalent dazu, dass ein solcher Graph eine Bucheinbettung auf zwei Seiten hat. Der Beweis ist konstruktiv und zeigt in der Tat sogar, dass es für die Realisierbarkeit nur notwendig ist, den Grad von Knoten separierender 3-Kreise zu beschränken - die übrigen Knoten können beliebig hohe Grade aufweisen. Dieses Ergebnis ist bestmöglich: Wenn die Gradschranke auf 6 abgeschwächt wird, gibt es Gegenbeispiele. Diese Ergebnisse verbessern Resultate von Heath [1995] und von Bauernöppel [1987] und, unabhängig davon, Bekos, Gronemann und Raftopoulou [2016], die gezeigt haben, dass planare Graphen mit Maximalgrad 3 beziehungsweise 4 auf zwei Seiten realisiert werden können. (3) Konvexitätssteigernde Deformationen: Wir zeigen, dass jede planare geradlinige Zeichnung eines intern dreifach zusammenhängenden planaren Graphen stetig zu einer solchen deformiert werden kann, in der jede Fläche ein konvexes Polygon ist. Dabei erhält die Deformation die Planarität und ist konvexitätssteigernd - sobald ein Winkel konvex ist, bleibt er konvex. Wir geben einen effizienten Algorithmus an, der eine solche Deformation berechnet, die aus einer asymptotisch optimalen Anzahl von Schritten besteht. In jedem Schritt bewegen sich entweder alle Knoten entlang horizontaler oder entlang vertikaler Geraden

Packing and embedding large subgraphs

This thesis contains several embedding results for graphs in both random and non random settings. Most notably, we resolve a long standing conjecture that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals $1/2$. %posed e.g.~by Bollob\'as, In Chapter 2 we obtain the following perturbation result regarding the hypercube \cQ^n: if H\subseteq\cQ^n satisfies $\delta(H)\geq\alpha n$ with $\alpha>0$ fixed and we consider a random binomial subgraph \cQ^n_p of \cQ^n with $p\in(0,1]$ fixed, then with high probability H\cup\cQ^n_p contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\in\mathbb{N}$. This result is part of a larger volume of work where we also prove the corresponding hitting time result for Hamiltonicity. In Chapter 3 we move to a non random setting. %to a deterministic one. %Instead of embedding a single Hamilton cycle our result concerns packing more general families of graphs into a fixed host graph. Rather than pack a small number of Hamilton cycles into a fixed host graph, our aim is to achieve optimally sized packings of more general families of graphs. More specifically, we provide a degree condition on a regular $n$-vertex graph $G$ which ensures the existence of a near optimal packing of any family $\mathcal H$ of bounded degree $n$-vertex $k$-chromatic separable graphs into $G$. %In general, this degree condition is best possible. %In particular, this yields an approximate version of the tree packing conjecture %in the setting of regular host graphs $G$ of high degree. %Similarly, our result implies approximate versions of the Oberwolfach problem, %the Alspach problem and the existence of resolvable designs in the setting of %regular host graphs of high degree. In particular, this yields approximate versions of the the tree packing conjecture, the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree

Hypercube-Based Topologies With Incremental Link Redundancy.

Hypercube structures have received a great deal of attention due to the attractive properties inherent to their topology. Parallel algorithms targeted at this topology can be partitioned into many tasks, each of which running on one node processor. A high degree of performance is achievable by running every task individually and concurrently on each node processor available in the hypercube. Nevertheless, the performance can be greatly degraded if the node processors spend much time just communicating with one another. The goal in designing hypercubes is, therefore, to achieve a high ratio of computation time to communication time. The dissertation addresses primarily ways to enhance system performance by minimizing the communication time among processors. The need for improving the performance of hypercube networks is clearly explained. Three novel topologies related to hypercubes with improved performance are proposed and analyzed. Firstly, the Bridged Hypercube (BHC) is introduced. It is shown that this design is remarkably more efficient and cost-effective than the standard hypercube due to its low diameter. Basic routing algorithms such as one to one and broadcasting are developed for the BHC and proven optimal. Shortcomings of the BHC such as its asymmetry and limited application are clearly discussed. The Folded Hypercube (FHC), a symmetric network with low diameter and low degree of the node, is introduced. This new topology is shown to support highly efficient communications among the processors. For the FHC, optimal routing algorithms are developed and proven to be remarkably more efficient than those of the conventional hypercube. For both BHC and FHC, network parameters such as average distance, message traffic density, and communication delay are derived and comparatively analyzed. Lastly, to enhance the fault tolerance of the hypercube, a new design called Fault Tolerant Hypercube (FTH) is proposed. The FTH is shown to exhibit a graceful degradation in performance with the existence of faults. Probabilistic models based on Markov chain are employed to characterize the fault tolerance of the FTH. The results are verified by Monte Carlo simulation. The most attractive feature of all new topologies is the asymptotically zero overhead associated with them. The designs are simple and implementable. These designs can lead themselves to many parallel processing applications requiring high degree of performance

Entangled graphs on surfaces in space

In the chemical world, as well as the physical, strands get tangled. When those strands form loops, the mathematical discipline of ‘knot theory’ can be used to analyse and describe the resultant tangles. However less has been studied about the situation when the strands branch and form entangled loops in either finite structures or infinite periodic structures. The branches and loops within the structure form a ‘graph’, and can be described by mathematical ‘graph theory’, but when graph theory concerns itself with the way that a graph can fit in space, it typically focuses on the simplest ways of doing so. Graph theory thus provides few tools for understanding graphs that are entangled beyond their simplest spatial configurations. This thesis explores this gap between knot theory and graph theory. It is focussed on the introduction of small amounts of entanglement into finite graphs embedded in space. These graphs are located on surfaces in space, and the surface is chosen to allow a limited amount of complexity. As well as limiting the types of entanglement possible, the surface simplifies the analysis of the problem – reducing a three-dimensional problem to a two-dimensional one. Through much of this thesis, the embedding surface is a torus (the surface of a doughnut) and the graph embedded on the surface is the graph of a polyhedron. Polyhedral graphs can be embedded on a sphere, but the addition of the central hole of the torus allows a certain amount of freedom for the entanglement of the edges of the graph. Entanglements of the five Platonic polyhedra (tetrahedron, octahedron, cube, dodecahedron, icosahedron) are studied in depth through their embeddings on the torus. The structures that are produced in this way are analysed in terms of their component knots and links, as well as their symmetry and energy. It is then shown that all toroidally embedded tangled polyhedral graphs are necessarily chiral, which is an important property in biochemical and other systems. These finite tangled structures can also be used to make tangled infinite periodic nets; planar repeating subgraphs within the net can be systematically replaced with a tangled version, introducing a controlled level of entanglement into the net. Finally, the analysis of entangled structures simply in terms of knots and links is shown to be deficient, as a novel form of tangling can exist which involves neither knots nor links. This new form of entanglement is known as a ravel. Different types of ravels can be localised to the immediate vicinity of a vertex, or can be spread over an arbitrarily large scope within a finite graph or periodic net. These different forms of entanglement are relevant to chemical and biochemical self-assembly, including DNA nanotechnology and metal-ligand complex crystallisation

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Toric Topology

Toric topology emerged in the end of the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra. It has quickly grown up into a very active area with many interdisciplinary links and applications, and continues to attract experts from different fields. The key players in toric topology are moment-angle manifolds, a family of manifolds with torus actions defined in combinatorial terms. Their construction links to combinatorial geometry and algebraic geometry of toric varieties via the related notion of a quasitoric manifold. Discovery of remarkable geometric structures on moment-angle manifolds led to seminal connections with the classical and modern areas of symplectic, Lagrangian and non-Kaehler complex geometry. A related categorical construction of moment-angle complexes and their generalisations, polyhedral products, provides a universal framework for many fundamental constructions of homotopical topology. The study of polyhedral products is now evolving into a separate area of homotopy theory, with strong links to other areas of toric topology. A new perspective on torus action has also contributed to the development of classical areas of algebraic topology, such as complex cobordism. The book contains lots of open problems and is addressed to experts interested in new ideas linking all the subjects involved, as well as to graduate students and young researchers ready to enter into a beautiful new area.Comment: Preliminary version. Contains 9 chapters, 5 appendices, bibliography, index. 495 pages. Comments and suggestions are very welcom

High Performance Software Reconfiguration in the Context of Distributed Systems and Interconnection Networks.

Designed algorithms that are useful for developing protocols and supporting tools for fault tolerance, dynamic load balancing, and distributing monitoring in loosely coupled multi-processor systems. Four efficient algorithms are developed to learn network topology and reconfigure distributed application programs in execution using the available tools for replication and process migration. The first algorithm provides techniques for transparent software reconfiguration based on process migration in the context of quadtree embeddings in Hypercubes. Our novel approach provides efficient reconfiguration for some classes of faults that may be identified easily. We provide a theoretical characterization to use graph matching, quadratic assignment, and a variety of branch and bound techniques to recover from general faults at run-time and maintain load balance. The second algorithm provides distributed recognition of articulation points, biconnected components, and bridges. Since the removal of an articulation point disconnects the network, knowledge about it may be used for selective replication. We have obtained the most efficient distributed algorithms with linear message complexity for the recognition of these properties. The third algorithm is an optimal linear message complexity distributed solution for recognizing graph planarity which is one of the most celebrated problems in graph theory and algorithm design. Recently, efficient shortest path algorithms are developed for planar graphs whose efficient recognition itself was left open. Our algorithm also leads to designing efficient distributed algorithm to recognize outer-planar graphs with applications in Hamiltonian path, shortest path routing and graph coloring. It is shown that efficient routing of information and distributing the stack needed for for planarity testing permit local computations leading to an efficient distributed algorithm. The fourth algorithm provides software redundancy techniques to provide fault tolerance to program structures. We consider the problem of mapping replicated program structures to provide efficient communication between modules in multiple replicas. We have obtained an optimal mapping of 2-replicated binary trees into hypercubes. For replication numbers greater than two, we provide efficient heuristic simulation results to provide efficient support for both \u27N-version programming\u27 and \u27Recovery block\u27 approaches for software replication

Stochastic Analysis

[no abstract available