Rok Požar (2013) *Complexity and algorithmic aspects in the theory of graph covers*. PhD thesis.

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## Abstract

Apart from mathematics, covering techniques have long been known as a powerful tool in different areas of science, especially in those fields dealing with representation and analysis of large structural objects. In such a context, one of the key properties of graph covers is that all the information about a (usually) large covering graph can be encoded by means of voltages assigned to directed edges of a (relatively) small base graph. In addition, the study of structural properties of covering graphs often reduces to the study of voltage distribution on the base graph. Combinatorial techniques developed for this purpose are based on the concept of lifting automorphism along covering projections. The aim of this Thesis is twofold. Firstly, to develop adequate algorithms for solving certain natural problems regarding lifting automorphisms. And secondly, to provide an appropriate software package for practical usage. An efficient algorithm for testing whether a given automorphism lifts along a covering projection, given in terms of voltages, is presented. Further, algorithms for analysing the structure of lifted groups along combinatorially given regular covering projection are developed: an algorithm for finding a presentation of the lifted group, an algorithm for testing whether the lifted group is a split extension, and algorithm for testing whether the lifted group is a sectional split extension. All algorithms avoid explicit constructions of the covering graph as well as of the lifted group, since such constructions are time and space consuming in general. In addition, methods for generating those covering projections along which a given group of automorphisms lifts, are presented. In particular, a method for finding regular covering projections along which a given group of automorphisms lifts as a sectional split extension, is given.

Item Type: | Thesis (PhD thesis) | |||||||||
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Keywords: | Apart from mathematics, covering techniques have long been known as a powerful tool in different areas of science, especially in those fields dealing with representation and analysis of large structural objects. In such a context, one of the key properties of graph covers is that all the information about a (usually) large covering graph can be encoded by means of voltages assigned to directed edges of a (relatively) small base graph. In addition, the study of structural properties of covering graphs often reduces to the study of voltage distribution on the base graph. Combinatorial techniques developed for this purpose are based on the concept of lifting automorphism along covering projections. The aim of this Thesis is twofold. Firstly, to develop adequate algorithms for solving certain natural problems regarding lifting automorphisms. And secondly, to provide an appropriate software package for practical usage. An efficient algorithm for testing whether a given automorphism lifts along a covering projection, given in terms of voltages, is presented. Further, algorithms for analysing the structure of lifted groups along combinatorially given regular covering projection are developed: an algorithm for finding a presentation of the lifted group, an algorithm for testing whether the lifted group is a split extension, and algorithm for testing whether the lifted group is a sectional split extension. All algorithms avoid explicit constructions of the covering graph as well as of the lifted group, since such constructions are time and space consuming in general. In addition, methods for generating those covering projections along which a given group of automorphisms lifts, are presented. In particular, a method for finding regular covering projections along which a given group of automorphisms lifts as a sectional split extension, is given. | |||||||||

Number of Pages: | 128 | |||||||||

Language of Content: | Slovenian | |||||||||

Mentor / Comentors: |
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Link to COBISS: | http://www.cobiss.si/scripts/cobiss?command=search&base=50070&select=(ID=10193492) | |||||||||

Institution: | University of Ljubljana | |||||||||

Department: | Faculty of Computer and Information Science | |||||||||

Item ID: | 2203 | |||||||||

Date Deposited: | 23 Sep 2013 10:04 | |||||||||

Last Modified: | 15 Oct 2013 09:43 | |||||||||

URI: | http://eprints.fri.uni-lj.si/id/eprint/2203 |

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