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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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Extremal combinatorics of matrices, sequences and sets of permutations
Cibulka, Josef ; Valtr, Pavel (advisor) ; Füredi, Zoltán (referee) ; Jelínek, Vít (referee)
Title: Extremal combinatorics of matrices, sequences and sets of permutations Author: Josef Cibulka Department: Department of Applied Mathematics Supervisor: Doc. RNDr. Pavel Valtr, Dr., Department of Applied Mathematics Abstract: This thesis studies questions from the areas of the extremal theory of {0, 1}-matrices, sequences and sets of permutations, which found many ap- plications in combinatorial and computational geometry. The VC-dimension of a set P of n-element permutations is the largest integer k such that the set of restrictions of the permutations in P on some k-tuple of positions is the set of all k! permutation patterns. We show lower and upper bounds quasiexponential in n on the maximum size of a set of n-element permutations with VC-dimension bounded by a constant. This is used in a paper of Jan Kynčl to considerably improve the upper bound on the number of weak isomorphism classes of com- plete topological graphs on n vertices. For some, mostly permutation, matrices M, we give new bounds on the number of 1-entries an n × n M-avoiding matrix can have. For example, for every even k, we give a construction of a matrix with k2 n/2 1-entries that avoids one specific k-permutation matrix. We also give almost tight bounds on the maximum number of 1-entries in matrices avoiding a fixed layered...
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Intersection representations of graphs
Töpfer, Martin ; Jelínek, Vít (advisor) ; Pangrác, Ondřej (referee)
This thesis is devoted to the outer and grounded string representations of graphs and their subclasses. A string representation of a graph is a set of strings (bounded continuous curves in a plane), where each string corresponds to one vertex of the graph. Two strings intersect each other if and only if the two corresponding vertices are adjacent in the original graph. An outer string graph is a graph with a string representation where strings are realized inside a disk and one endpoint of each string lies on the boundary of the disk. Similarly, in case of grounded string graphs the strings lie in a common half- plane with one endpoint of each string on the boundary of the half-plane. We give a summary of subclasses of grounded string graphs and proves several results about their mutual inclusions and separations. To prove those, we use an order-forcing lemma which can be used to force a particular order of the endpoints of the string on the boundary circle or boundary line. The second part of the thesis contains proof that recognition of outer string graphs is NP-hard. 1
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Partial representation extension for subclasses of interval graphs
Onduš, Daniel ; Kratochvíl, Jan (advisor) ; Jelínek, Vít (referee)
The problem of extending partial representations for an interval graph asks, whether it is possible to extend a given representation of some vertices to a valid representation of the entire graph. In this thesis we extend the recent result of Klavík et al. who proved REPEXT can be decided for proper and unit interval graphs in polynomial time. We describe properties of PI± and U± graphs and their representations and present algorithms deciding REPEXT for these classes in polynomial time. In the process, we characterize relations between the K1,3's in a graph and show that we can decide the open vertex of every K1,3. We also define notions of representation of the same order type and locally similar representations as well as intervals forced and locally forced to be closed (open) that are essential for extending partial representations when multiple types of intervals can occur in the same representation. We characterize intervals forced and locally forced to be closed (open) in a U± graph using integer gaps in the pre-representation and we construct lower bounds for the rightmost endpoint of a component in polynomial time.
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Matrices without forbidden interval minors
Surma, David ; Jelínek, Vít (advisor) ; Klazar, Martin (referee)
In the thesis, we study the structure of binary matrices which do not contain a pat- tern P as an interval minor. We also deal with matrices that are critical for P, i.e., matrices without P which after changing any 0-entry to 1-entry contain the forbidden pattern P. First, we describe matrices critical for any one-line pattern. Then we deal with all patterns with two rows and three columns which contain at most four 1-entries. Finally, we characterize the matrices critical for the alternating pattern of size 2 × 4. 1
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Algorithmic aspects of intersection representations
Chmel, Petr ; Jelínek, Vít (advisor) ; Kratochvíl, Jan (referee)
As some problems are (NP-)hard to solve in the general case, a possible approach is to try to solve the problem on a restricted class of graphs. In the thesis, we focus on graphs induced by axis-aligned L-shapes, so-called L-graphs, and a similar class of axis- aligned L-shapes and L-shapes, referred to as {L, L}-graphs, with two vertices sharing an edge if and only if their respective curves intersect. We show that recognizing both L- graphs and {L, L}-graphs is NP-complete. The second part of the thesis focuses on other typical decision problems on L-graphs and their relatives: finding the clique number, the independence number or a 3-coloring.
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