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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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Algorithmic aspects of intersection-defined graph classes
Jedličková, Nikola ; Kratochvíl, Jan (advisor) ; Fiala, Jiří (referee)
Geometrically representable graphs are extensively studied area of research in contempo- rary literature due to their structural characterizations and efficient algorithms. The most frequently studied class of such graphs is the class of interval graphs. In this thesis we focus on two problems, generalizing the problem of recognition, for classes related to interval graphs. In the first part, we are concerned with adjusted interval graphs. This class has been studied as the right digraph analogue of interval graphs. For interval graphs, there are polynomial algorithms to extend a partial representation by given intervals into a full interval representation. We will introduce a similar problem - the partial ordering extension - and we will provide a polynomial algorithm to extend a partial ordering of adjusted interval digraphs. In the second part, we show two NP-completeness results regarding the simultaneous representation problem, introduced by Lubiw and Jampani. The simultaneous representation problem for a given class of intersection graphs asks if some k graphs can be represented so that every vertex is represented by the same object in each representation. We prove that it is NP-complete to decide this for the class of interval and circular-arc graphs in the case when k is a part of the input and graphs...
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Intersection representations of graphs
Töpfer, Martin ; Jelínek, Vít (advisor) ; Šámal, Robert (referee)
We investigate intersection graphs of axis-aligned L-shapes (L-graphs) and their subclass, when all L- shapes have one of their endpoints on the same line - let us call this class outer-L-graphs. In the beginning we introduce some known facts about L-graphs. Then we show, that interval, circular and outerplanar graphs are subclasses of outer-L-graphs. We also characterise outer-L-graphs using ordering without forbidden patterns. Powered by TCPDF (www.tcpdf.org)
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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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Algoritmické problémy související s průnikovými grafy
Ivánek, Jindřich ; Pergel, Martin (advisor) ; Rytíř, Pavel (referee)
In this thesis we study two clique-cover problems which have interesting applications regarding the k -bend intersection graph representation: the edge-clique-cover-degree problem and the edge-clique-layered-cover problem. We focus on the complexity of these problems and polynomial time algorithms on restricted classes of graphs. The main results of the thesis are NP-completness of the edge-clique-layered-cover problem and a polynomial-time 2-approximation algorithm on the subclass of diamond-free graphs for the same problem as well as some upper bounds on particular graph classes.
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