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On-line algorithms for bipartite graph coloring
Chludil, Josef ; Pangrác, Ondřej (advisor) ; Gavenčiak, Tomáš (referee)
Instance of the on-line graph coloring problem is a graph together with a permutation of its vertices (viewed as a linear ordering of the vertex set). The goal is to color the graph with vertices taken in the given order using the information of the subgraph induced by previous vertices. The most natural algorithm is the First Fit algorithm using in each step the first possible color. Unfortunately optimum number of colors can be linear dependent on number of vertices of graph even for bipartite graphs. On the other hand, there is an algorithm with logarithmic approximation factor for this class of graphs.
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Hry na grafech
Gavenčiak, Tomáš ; Kratochvíl, Jan (advisor) ; Pergel, Martin (referee)
In this thesis we study properties of one cop&robber game. In this game two players (Cop and Robber) take turns in moving on a finite undirected graph. Both players move with the speed at most one edge per turn. They both know the complete game status. If at any time Cop shares a vertex with Robber, Cop wins. If that never happens, Robber wins. Games of this type are important as models of searching in graphs and networks and for the connection to the width parameters of graphs. We closely examine the class of graphs with a winning strategy for Cop (the so called cop-win graphs) and construct best strategies for both Cop and Robber. The previously known results include the fact that the number of moves in which Cop can catch Robber on every cop-win graph on n vertices is bounded by n3 and there are graphs which require n 4. We show that this number is exactly n 4.
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Structural and complexity questions of graph theory
Gavenčiak, Tomáš ; Kratochvíl, Jan (advisor) ; Hliněný, Petr (referee) ; Fomin, Fedor (referee)
DOCTORAL THESIS ABSTRACT Tomáš Gavenčiak Structural and complexity questions of graph theory The original cops and robber game, proposed in 1983 by Nowakowski and Winkler, and independently by Quilliot, is a two-player combinatorial pursuit game on a graph. Many related games have been introduced and studied since then, both with a purely game theoretic motivation and for their applications in graph theory and connections with graph width parameters. We give an overview of the field and present three particular results: We show bounds on the required number of cops for various connected intersection graph classes, most notably we show that on connected string graphs 15 cops always win. We show the game of cops and fast robber to be polynomially decidable on interval graphs and as a tool we generalise the problem of defensive domination and show a polynomial algorithm for interval graphs. Finally we propose and examine generalisations of tree-depth, marshals and robber games, and minors to hypergraphs and hypergraph pairs.
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Hry na grafech ve vztahu k zdvihovým parametrům grafů
Gavenčiak, Tomáš ; Kratochvíl, Jan (advisor) ; Smrž, Otakar (referee)
We consider a variant of a cop and robber game with an in nitely fast robber and its relations to other similar games. We compare the helicopter game characterizing tree-width, the classical cop and robber game and its versions with various speeds of the robber. We study the complexity of the in nitely fast robber variant and give an explicit characterization of all the graphs where one cop can win. As the main result, we show a polynomial time algorithm deciding the game on interval graphs. This answers a question from the paper Fomin et al.: On tractability of Cops and Robbers game, IFIP TCS 2008, 171-185. To show the polynomiality of the game on interval graphs, we introduce a new auxiliary game on an interval representation of the graph and show the polynomiality of that game. Then we use game strategy reductions to show the equivalence of the two games.
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Compact I/O-Efficient Graph Representations
Tětek, Jakub ; Gavenčiak, Tomáš (advisor) ; Mareš, Martin (referee)
The objective of this thesis is to develop a fast memory-efficient representa- tion of some graphs that occur in real-world applications. We consider separable graph classes (e.g. planar graphs or graphs of bounded genus) and show how to represent them in a way that (1) makes accessing vertices in a walk cache-efficient on average and (2) is highly memory-efficient. In particular, we show a compact representation of separable graph classes with the I/O cost of a random walk of length k being O(K/(Bw)1−c ) w.h.p. In the second part of the thesis, we consider layout of trees with optimal worst-case I/O cost for root-to-leaf traversal, show an additive (+1)-approximation of I/O optimal compact layout and contrast this with a proof of NP-hardness of exact solution. In this thesis, we also prove generalisations of the recursive separator theo- rem. The first one generalises the theorem for weighted graphs and the second one replaces minimum region size by average region size in the bound. 1
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Routing information exchange
Čepelík, David ; Mareš, Martin (advisor) ; Gavenčiak, Tomáš (referee)
We present an efficient method for serialization of complex objects into a compact binary form based on the Concise Binary Object Representation (CBOR). We provide a high-performance, well-tested imple- mentation of a data serialization library and implement it in the BIRD Internet Routing Daemon. 1
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Structural and complexity questions of graph theory
Gavenčiak, Tomáš ; Kratochvíl, Jan (advisor) ; Hliněný, Petr (referee) ; Fomin, Fedor (referee)
DOCTORAL THESIS ABSTRACT Tomáš Gavenčiak Structural and complexity questions of graph theory The original cops and robber game, proposed in 1983 by Nowakowski and Winkler, and independently by Quilliot, is a two-player combinatorial pursuit game on a graph. Many related games have been introduced and studied since then, both with a purely game theoretic motivation and for their applications in graph theory and connections with graph width parameters. We give an overview of the field and present three particular results: We show bounds on the required number of cops for various connected intersection graph classes, most notably we show that on connected string graphs 15 cops always win. We show the game of cops and fast robber to be polynomially decidable on interval graphs and as a tool we generalise the problem of defensive domination and show a polynomial algorithm for interval graphs. Finally we propose and examine generalisations of tree-depth, marshals and robber games, and minors to hypergraphs and hypergraph pairs.
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Optimisation using graph searching on special graph classes
Chejnovská, Anna ; Gavenčiak, Tomáš (advisor) ; Šámal, Robert (referee)
For a given graph we define the minimum path cover as a minimum cardinality set of vertex disjoint paths covering all the vertices of the graph. This problem is one of the usual generalization of the Hamiltonian path problem. In this thesis we based our work on a paper Corneil et al. (2013) presenting a certifying algorithm for the minimal path cover problem on cocomparability graphs (the complement of graph of strict partial order). We first introduce this algorithm an then we experimentally examine its robustness to five operations on edges and vertices of the graph. We also analyse the impact of these operation on the size of the minimal path cover theoretically. Powered by TCPDF (www.tcpdf.org)
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Cops and robber game on directed complete graphs
Slívová, Veronika ; Gavenčiak, Tomáš (advisor) ; Jelínek, Vít (referee)
BACHELOR THESIS - ABSTRACT Veronika Slívová This thesis focuses on the game Cops and robber on tournaments (a graph we gain by orienting each edge of undirected complete graph). We show that tournaments containing a vertex with large outdegree have small cop number. On the other hand we prove that the number of cops we need to capture the robber on any tournament is not bounded. Then we study circulant tournaments and graphs obtained by cyclically orienting each triple of Steiner triple system. We disprove the conjecture made by Geňa Hahn that the cop number of any graph obtained by orientation Steiner triple system is bounded. We also show that the cop number of circulant tournaments cannot be bounded. Then we focus on the 2-fast cop and the 2-fast robber variant of the game. We prove that one 2-fast cop wins on any tournament. On the contrary in case that the game is played on tournament whose vertices have the same outdegree, the 2-fast robber is captured trivially or runs away indefinitely.
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