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Groups of automorphisms of graphs
Zeman, Peter ; Nedela, Roman (advisor) ; Felsner, Stefan (referee) ; Širáň, Jozef (referee)
Groups of automorphisms of graphs - abstract In this thesis we investigate automorphism groups of several restricted classes of graphs from structural and computational point of view. For interval, permutation, circle, and planar graphs we give inductive characterizations of their automorphism groups in terms of group products. For chordal graphs of bounded leafage, prove that computing the automorphism group, and consequently the isomorphism problem, is fixed parameter tractable. For maps on surfaces, we give a linear time algorithm computing the automor- phism group, parametrized by the genus of the underlying surface. 1
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Regulární nakrytí - struktura a složitost
Seifrtová, Michaela ; Fiala, Jiří (advisor) ; Nedela, Roman (referee)
Regular Coverings - Structure and Complexity Michaela Seifrtová The thesis consists of two main parts, the first concentrated on the struc- ture of graph coverings, where different properties of regular graph coverings are presented, and the second dealing with computational complexity of the covering problem. Favorable results have been achieved in this area, proving the problem is solvable in polynomial time for all graphs whose order is a prime multiple of the order of the covered graph. 1
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Automorphism Groups of Geometrically Represented Graphs
Zeman, Peter ; Klavík, Pavel (advisor) ; Nedela, Roman (referee)
In this thesis, we are interested in automorphism groups of classes of graphs with a very strong structure. Probably the first nontrivial result in this direction is from 1869 due to Jordan. He gave a characterization of the class T of the automorphism groups of trees. Surprisingly, automorphism groups of intersection-defined classes of graphs were studied only briefly. Even for deeply studied classes of intersection graphs the structure of their automorphism groups is not well known. We study the problem of reconstruct- ing the automorphism group of a geometric intersection graph from a good knowledge of the structure of its representations. We mainly deal with interval graphs. Interval graphs are intersection graphs of intervals on the real line. They are one of the oldest and most studied classes of geometric intersection graphs. Our main result is that the class T is the same as the class I of the automorphism groups of interval graphs. Moreover, we show for an interval graph how to find a tree with the same automorphism group, and vice versa. 1
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Additive combinatorics and number theory
Hančl, Jaroslav ; Klazar, Martin (advisor) ; Balogh, Jozsef (referee) ; Nedela, Roman (referee)
We present several results for growth functions of ideals of different com- binatorial structures. An ideal is a set downward closed under a containment relation, like the relation of subpartition for partitions, or the relation of induced subgraph for graphs etc. Its growth function (GF) counts elements of given size. For partition ideals we establish an asymptotics for GF of ideals that do not use parts from a finite set S and use this to construct ideal with highly oscillating GF. Then we present application characterising GF of particular partition ideals. We generalize ideals of ordered graphs to ordered uniform hypergraphs and show two dichotomies for their GF. The first result is a constant to linear jump for k-uniform hypergraphs. The second result establishes the polynomial to exponential jump for 3-uniform hypergraphs. That is, there are no ordered hypergraph ideals with GF strictly inside the constant-linear and polynomial- exponential range. We obtain in both dichotomies tight upper bounds. Finally, in a quite general setting we present several methods how to generate for various combinatorial structures pairs of sets defining two ideals with iden- tical GF. We call these pairs Wilf equivalent pairs and use the automorphism method and the replacement method to obtain such pairs. 1
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Automorphism Groups of Geometrically Represented Graphs
Zeman, Peter ; Klavík, Pavel (advisor) ; Nedela, Roman (referee)
In this thesis, we are interested in automorphism groups of classes of graphs with a very strong structure. Probably the first nontrivial result in this direction is from 1869 due to Jordan. He gave a characterization of the class T of the automorphism groups of trees. Surprisingly, automorphism groups of intersection-defined classes of graphs were studied only briefly. Even for deeply studied classes of intersection graphs the structure of their automorphism groups is not well known. We study the problem of reconstruct- ing the automorphism group of a geometric intersection graph from a good knowledge of the structure of its representations. We mainly deal with interval graphs. Interval graphs are intersection graphs of intervals on the real line. They are one of the oldest and most studied classes of geometric intersection graphs. Our main result is that the class T is the same as the class I of the automorphism groups of interval graphs. Moreover, we show for an interval graph how to find a tree with the same automorphism group, and vice versa. 1
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Regulární nakrytí - struktura a složitost
Seifrtová, Michaela ; Fiala, Jiří (advisor) ; Nedela, Roman (referee)
Regular Coverings - Structure and Complexity Michaela Seifrtová The thesis consists of two main parts, the first concentrated on the struc- ture of graph coverings, where different properties of regular graph coverings are presented, and the second dealing with computational complexity of the covering problem. Favorable results have been achieved in this area, proving the problem is solvable in polynomial time for all graphs whose order is a prime multiple of the order of the covered graph. 1
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