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Structural and Algorithmic Properties of Permutation Classes
Opler, Michal ; Jelínek, Vít (advisor) ; Kozma, László (referee) ; Bouvel, Mathilde (referee)
In this thesis, we study the relationship between the structure of permutation classes and the computational complexity of different decision problems. First, we explore the structure of permutation classes through the lens of various parameters, with a particular interest in tree-width. We define novel structural properties of a general permutation class C, the most notable being the long path property. Using these properties, we infer lower bounds on the maximum tree-width attained by a permutation of length n in C. For example, we prove that any class with the long path property has unbounded tree-width. The main decision problem we consider is known as Permutation Pattern Match- ing (PPM). The input of PPM consists of a pair of permutations τ (the 'text') and π (the 'pattern'), and the goal is to decide whether τ contains π as a subpermutation. Af- ter briefly considering general PPM, we focus on its pattern-restricted variant known as C-Pattern PPM where we additionally require that the pattern π comes from a fixed class C. We derive both classical and parameterized hardness results assuming different structural properties of C. For example, we show that C-Pattern PPM is NP-complete whenever C has the long path property. Furthermore, we focus on an even more restricted variant of PPM where the text is...
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The combinatorics of pattern-avoiding matrices
Mikšaník, David ; Jelínek, Vít (advisor) ; Klazar, Martin (referee)
A permutation matrix P partially avoids a quasi-permutation matrix A (i.e., a 01- matrix such that each column and row of A contains at most one nonzero entry) if there is no submatrix P′ of P of the same size as A satisfying Ai,j ≤ P′ i,j for all possible indices i and j. Two quasi-permutation matrices A and B are partially Wilf-equivalent if, for every n ∈ N, the number of permutation matrices of order n partially avoiding A is the same as the number of permutation matrices of order n partially avoiding B. This generalizes the well-known concept of avoidance and Wilf equivalence of permutations. One of the central topics in this area is the classification of permutations of order k into Wilf equivalence classes. The complete classification is known for k = 1, 2, . . . , 7. In our thesis, we study the same problem for quasi-permutation matrices. Namely, we classify all 371 quasi-permutation matrices of size at most 4 × 4 into partial Wilf equivalence classes (two quasi-permutation matrices belong to the same class if and only if they are partially Wilf-equivalent). Along the way, we prove several general results showing how to construct from one or two quasi-permutation matrices more quasi-permutation matrices that are pairwise partially Wilf-equivalent. 1
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New Intersection Graph Hierachies
Chmel, Petr ; Jelínek, Vít (advisor) ; Kratochvíl, Jan (referee)
String graphs are the intersection graphs of curves in the plane. Asinowski et al. [JGAA 2012] introduced a hierarchy of VPG graphs based on the number of bends and showed that the hierarchy contains precisely all string graphs. A similar hierarchy can be observed with k-string graphs: string graphs with the additional condition that each pair of curves has at most k intersection points. We continue in this direction by introducing precisely-k-string graphs which restrict the representation even more so that each pair of curves has either 0 or precisely k intersection points with all of them being crossings. We prove that for each k ≥ 1, any precisely-k-string graph is a precisely-(k + 2)-string graph and that the classes of precisely-k-string graphs and precisely-(k + 1)-string graphs are incomparable with respect to inclusion. We also investigate the problem of finding an efficiently representable class of intersec- tion graphs of objects in the plane that contains all graphs with fixed maximum degree. In the process, we introduce a new hierarchy of intersection graphs of unions of d hori- zontal or vertical line segments, called impure-d-line graphs, and other variations of the class with representation restrictions. We prove that all graphs with maximum degree ≤ 2d are impure-d-line graphs and for d = 1...
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The Möbius function of combinatorial posets
Kopfová, Lenka ; Jelínek, Vít (advisor) ; Kantor, Ida (referee)
In this thesis we study the poset of signed permutations under the pattern containment order. A signed permutation is a permutation in which each entry has a plus or a minus sign assigned to it. Therefore signed permutations are a generalization of unsigned permutations as those correspond to picking the plus sign for each entry. We present several results regarding the M¨obius function of signed permutations, some of which are generalizations of those for unsigned ones. Moreover, we study the poset isomorphism between intervals of the poset of signed permutations, which ensures that two intervals have the same value of the M¨obius function.
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Algebraické vlastnosti barevnosti grafů
Bulánek, Jan ; Kráľ, Daniel (advisor) ; Jelínek, Vít (referee)
We study algebraic tools for proving the existence of graph colorings and focus a classical method of Alon-Tarsi from this area. In particular, we prove Alon-Tarsi Theorem, provide examples of some known applications and give a new application to colorings of squares of cycles.
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Obecná enumerace číselných rozkladů
Hančl, Jaroslav ; Klazar, Martin (advisor) ; Jelínek, Vít (referee)
Název práce: Obecná enumerace číselných rozklad· Autor: Jaroslav Hančl Katedra: Katedra aplikované matematiky Vedoucí diplomové práce: doc. RNDr. Martin Klazar, Dr., KAM MFF UK Abstrakt: Předložená diplomová práce se zabývá asymptotikami počítacích funkcí ideál· číselných rozklad·. Jejím hlavním cílem je zjistit největší možný asympto- tický r·st počítací funkce rozkladového ideálu, která je nekonečněkrát rovna nule. Autor se na základě znalosti asymptotik vybraných rozkladových ideál· snaží po- mocí kombinatorických a základních analytických metod odvodit odhady hledané asymptotiky. Výsledkem je za prvé slabší horní odhad, za druhé poměrně silný dolní odhad a za třetí, pro speciální třídu rozkladových ideál· je nalezen největší asymptotický r·st. Klíčová slova: íselné rozklady, asymptotika rozklad·, rozkladové ideály, počítací funkce, kombinatorická enumerace. 1
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Generating random pattern-avoiding matrices
Kučera, Stanislav ; Jelínek, Vít (advisor) ; Šámal, Robert (referee)
Binary matrices not containing a smaller matrix as a submatrix have become an interesting topic recently. In my thesis, I introduce two new algorithms to test whether a big square binary matrix contains a smaller binary matrix together with a process using randomness, which approximates a uniformly random matrix not containing a given matrix. The reason to create such algorithms is to allow researchers test their conjectures on random matrices. Thus, my thesis also contains an effective cross- platform implementation of all mentioned algorithms. Powered by TCPDF (www.tcpdf.org)
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Generating graphs
Mohelníková, Lucie ; Dvořák, Zdeněk (advisor) ; Jelínek, Vít (referee)
Title: Generating graphs Author: Lucie Mohelníková Department: Department of Applied Mathematics Supervisor: Mgr. Zdeněk Dvořák, Ph.D., Computer Science Institute of Char- les University Abstract: The main topic of this thesis are the methods used to generate graphs from prescribed classes, especially graphs embeddable in surfaces. An im- portant technique in this context is to generate the graphs by vertex decontracti- ons. The identification of initial (irreducible) graphs is crucial for this technique. We give an overview of the results regarding the irreducible triangulations and quadrangulations of various surfaces, especially the surfaces of low genus (sphere, projective plane, Klein bottle). The main result of this work is the identification 21 irreducible triangulations which proves the result of Lawrencenko without using of information technology. Keywords: irreducible, triangulations, torus
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Editor matematických výrazů
Holaň, David ; Jelínek, Vít (advisor) ; Lidický, Bernard (referee)
In the present work we study the concept and implementation of "MaEd for LATEX", a portable graphical user interface program for creating and editing of LATEX formulae. The program is designed to allow a beginner to create even complex formulae without any knowledge of underlying LATEX source code. The user can also import his own source code and the program doesn't make unnecessary changes to the imported source code, like removing comments and indentation. The concept describes how was the program's GUI designed. The LATEX commands available for creation of math formulae are described along with their syntax and level of support in the program. The work also analyzes structure of LATEX source code.
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