104 research outputs found
Colorings of graphs, digraphs, and hypergraphs
Brooks' Theorem ist eines der bekanntesten Resultate ĂŒber GraphenfĂ€rbungen: Sei G ein zusammenhĂ€ngender Graph mit Maximalgrad d. Ist G kein vollstĂ€ndiger Graph, so lassen sich die Ecken von G so mit d Farben fĂ€rben, dass zwei benachbarte Ecken unterschiedlich gefĂ€rbt sind. In der vorliegenden Arbeit liegt der Fokus auf Verallgemeinerungen von Brooks Theorem fĂŒr FĂ€rbungen von Hypergraphen und gerichteten Graphen. Eine FĂ€rbung eines Hypergraphen ist eine FĂ€rbung der Ecken so, dass keine Kante monochromatisch ist. Auf Hypergraphen erweitert wurde der Satz von Brooks von R.P. Jones. Im ersten Teil der Dissertation werden Möglichkeiten aufgezeigt, das Resultat von Jones weiter zu verallgemeinern. KernstĂŒck ist ein Zerlegungsresultat: Zu einem Hypergraphen H und einer Folge f=(f_1,âŠ,f_p) von Funktionen, welche von V(H) in die natĂŒrlichen Zahlen abbilden, wird untersucht, ob es eine Zerlegung von H in induzierte Unterhypergraphen H_1,âŠ,H_p derart gibt, dass jedes H_i strikt f_i-degeneriert ist. Dies bedeutet, dass jeder Unterhypergraph H_i' von H_i eine Ecke v enthĂ€lt, deren Grad in H_i' kleiner als f_i(v) ist. Es wird bewiesen, dass die Bedingung f_1(v)+âŠ+f_p(v) \geq d_H(v) fĂŒr alle v fast immer ausreichend fĂŒr die Existenz einer solchen Zerlegung ist und gezeigt, dass sich die AusnahmefĂ€lle gut charakterisieren lassen. Durch geeignete Wahl der Funktion f lassen sich viele bekannte Resultate ableiten, was im dritten Kapitel erörtert wird. Danach werden zwei weitere Verallgemeinerungen des Satzes von Jones bewiesen: Ein Theorem zu DP-FĂ€rbungen von Hypergraphen und ein Resultat, welches die chromatische Zahl eines Hypergraphen mit dessen maximalem lokalen Kantenzusammenhang verbindet. Der zweite Teil untersucht FĂ€rbungen gerichteter Graphen. Eine azyklische FĂ€rbung eines gerichteten Graphen ist eine FĂ€rbung der Eckenmenge des gerichteten Graphen sodass es keine monochromatischen gerichteten Kreise gibt. Auf dieses Konzept lassen sich viele klassische FĂ€rbungsresultate ĂŒbertragen. Dazu zĂ€hlt auch Brooks Theorem, wie von Mohar bewiesen wurde. Im siebten Kapitel werden DP-FĂ€rbungen gerichteter Graphen untersucht. Insbesondere erfolgt der Transfer von Mohars Theorem auf DP-FĂ€rbungen. Das darauffolgende Kapitel befasst sich mit kritischen gerichteten Graphen. Insbesondere werden Konstruktionen fĂŒr diese angegeben und die gerichtete Version des Satzes von HajĂłs bewiesen.Brooksâ Theorem is one of the most known results in graph coloring theory: Let G be a connected graph with maximum degree d >2. If G is not a complete graph, then there is a coloring of the vertices of G with d colors such that no two adjacent vertices get the same color. Based on Brooks' result, various research topics in graph coloring arose. Also, it became evident that Brooks' Theorem could be transferred to many other coloring-concepts. The present thesis puts its focus especially on two of those concepts: hypergraphs and digraphs. A coloring of a hypergraph H is a coloring of its vertices such that no edge is monochromatic. Brooks' Theorem for hypergraphs was obtained by R.P. Jones. In the first part of this thesis, we present several ways how to further extend Jones' theorem. The key element is a partition result, to which the second chapter is devoted. Given a hypergraph H and a sequence f=(f_1,âŠ,f_p) of functions, we examine if there is a partition of into induced subhypergraphs H_1,âŠ,H_p such that each of the H_i is strictly f_i-degenerate. This means that in each non-empty subhypergraph H_i' of H_i there is a vertex v having degree d_{H_i'}(v
Topics in graph colouring and extremal graph theory
In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let be a connected graph with vertices and maximum degree . Let denote the graph with vertex set all proper -colourings of and two -colourings are joined by an edge if they differ on the colour of exactly one vertex.
Our first main result states that has a unique non-trivial component with diameter . This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree.
A Kempe change is the operation of swapping some colours , of a component of the subgraph induced by vertices with colour or . Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all -colourings of a graph are Kempe equivalent unless is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007).
Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs.
Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees
Graph Theory
Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures
Decomposing and packing polygons / Dania el-Khechen.
In this thesis, we study three different problems in the field of computational geometry: the partitioning of a simple polygon into two congruent components, the partitioning of squares and rectangles into equal area components while minimizing the perimeter of the cuts, and the packing of the maximum number of squares in an orthogonal polygon. To solve the first problem, we present three polynomial time algorithms which given a simple polygon P partitions it, if possible, into two congruent and possibly nonsimple components P 1 and P 2 : an O ( n 2 log n ) time algorithm for properly congruent components and an O ( n 3 ) time algorithm for mirror congruent components. In our analysis of the second problem, we experimentally find new bounds on the optimal partitions of squares and rectangles into equal area components. The visualization of the best determined solutions allows us to conjecture some characteristics of a class of optimal solutions. Finally, for the third problem, we present three linear time algorithms for packing the maximum number of unit squares in three subclasses of orthogonal polygons: the staircase polygons, the pyramids and Manhattan skyline polygons. We also study a special case of the problem where the given orthogonal polygon has vertices with integer coordinates and the squares to pack are (2 {604} 2) squares. We model the latter problem with a binary integer program and we develop a system that produces and visualizes optimal solutions. The observation of such solutions aided us in proving some characteristics of a class of optimal solutions
Jigsaw: Scalable software-defined caches
Shared last-level caches, widely used in chip-multi-processors (CMPs), face two fundamental limitations. First, the latency and energy of shared caches degrade as the system scales up. Second, when multiple workloads share the CMP, they suffer from interference in shared cache accesses. Unfortunately, prior research addressing one issue either ignores or worsens the other: NUCA techniques reduce access latency but are prone to hotspots and interference, and cache partitioning techniques only provide isolation but do not reduce access latency.United States. Defense Advanced Research Projects Agency (DARPA PERFECT contract HR0011-13-2-0005)Quanta Computer (Firm
Time Protection: the Missing OS Abstraction
Timing channels enable data leakage that threatens the security of computer
systems, from cloud platforms to smartphones and browsers executing untrusted
third-party code. Preventing unauthorised information flow is a core duty of
the operating system, however, present OSes are unable to prevent timing
channels. We argue that OSes must provide time protection in addition to the
established memory protection. We examine the requirements of time protection,
present a design and its implementation in the seL4 microkernel, and evaluate
its efficacy as well as performance overhead on Arm and x86 processors
Data conditioning and display for Apollo prelaunch checkout - Test matrix technique
Test matrix technique for checkout and launching of Apollo-Saturn spacecraf
Boundary properties of graphs
A set of graphs may acquire various desirable properties, if we apply suitable restrictions
on the set. We investigate the following two questions: How far, exactly, must one restrict
the structure of a graph to obtain a certain interesting property? What kind of tools are
helpful to classify sets of graphs into those which satisfy a property and those that do not?
Equipped with a containment relation, a graph class is a special example of a partially
ordered set. We introduce the notion of a boundary ideal as a generalisation of a notion
introduced by Alekseev in 2003, to provide a tool to indicate whether a partially ordered set
satisfies a desirable property or not. This tool can give a complete characterisation of lower
ideals defined by a finite forbidden set, into those that satisfy the given property and to
those that do not. In the case of graphs, a lower ideal with respect to the induced subgraph
relation is known as a hereditary graph class.
We study three interrelated types of properties for hereditary graph classes: the existence
of an efficient solution to an algorithmic graph problem, the boundedness of the graph
parameter known as clique-width, and well-quasi-orderability by the induced subgraph relation.
It was shown by Courcelle, Makowsky and Rotics in 2000 that, for a graph class, boundedness
of clique-width immediately implies an efficient solution to a wide range of algorithmic
problems. This serves as one of the motivations to study clique-width. As for well-quasiorderability,
we conjecture that every hereditary graph class that is well-quasi-ordered by
the induced subgraph relation also has bounded clique-width.
We discover the first boundary classes for several algorithmic graph problems, including
the Hamiltonian cycle problem. We also give polynomial-time algorithms for the dominating
induced matching problem, for some restricted graph classes.
After discussing the special importance of bipartite graphs in the study of clique-width,
we describe a general framework for constructing bipartite graphs of large clique-width. As
a consequence, we find a new minimal class of unbounded clique-width.
We prove numerous positive and negative results regarding the well-quasi-orderability of
classes of bipartite graphs. This completes a characterisation of the well-quasi-orderability of
all classes of bipartite graphs defined by one forbidden induced bipartite subgraph. We also
make considerable progress in characterising general graph classes defined by two forbidden
induced subgraphs, reducing the task to a small finite number of open cases. Finally, we
show that, in general, for hereditary graph classes defined by a forbidden set of bounded
finite size, a similar reduction is not usually possible, but the number of boundary classes
to determine well-quasi-orderability is nevertheless finite.
Our results, together with the notion of boundary ideals, are also relevant for the study
of other partially ordered sets in mathematics, such as permutations ordered by the pattern
containment relation
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