Predstavitve grafov z enotsko razdaljo

doktorska disertacija

Boris Horvat (Author), Franc Solina (Mentor), Tomaž Pisanski (Co-mentor)

Abstract

Disertacija opisuje probleme povezane z grafi, ki se jih da v evklidski ravnini predstaviti tako, da vozlišča predstavimo s točkami v ravnini povezave pa z daljicami dolžine ena. Probleme preučujemo tako z računalniškega (računskega) kot z matematičnega vidika. V uvodnem poglavju povzamemo do sedaj znane rezultate (predvsem matematične) teorije predstavitev grafov z enotsko razdaljo. Ob tem poenotimo terminologijo rezultatov, ki so nastajali v zadnjih petdesetih letih ter jo dopolnemo z dokazi nekaj manjših izrekov. Omenimo tudi prve poskuse generiranja majhnih grafov z enotsko razdaljo z računalnikom in predstavimo rezultate drugih avtorjev. V drugem poglavju obravnavamo najpomembnejše grafovske produkte ▫$k$▫-razsežnih grafov z enotsko razdaljo. V tretjem poglavju ovržemo napačno domnevo, da Heawoodov graf ni graf z enotsko razdaljo. V četrtem poglavju naštejemo vse, tudi degenerirane predstavitve z enotsko razdaljo Petersenovega grafa v ravnini ter obravnavamo relacije med njimi. V petem poglavju opazujemo posplošene Petersenove grafe in ▫$I$▫-grafe. Dokažemo izrek o izomorfizmih ▫$I$▫-grafov in s tem obstoj predstavitve z enotsko razdaljo za veliko večino ▫$I$▫-grafov. Postavimo nekaj domnev, ki jih potrdimo s pomočjo računalnika za vse ▫$I$▫-grafe na največ 2000 vozliščih. V šestem poglavju se ukvarjamo s teorijo izračunljivosti in opazujemo težavnost problema obstoja degenerirane predstavitve grafov z enotsko razdaljo. Dokažemo, da sta odločitveni problem obstoja ▫$k$▫-razsežne degenerirane predstavitve z enotsko razdaljo in odločitveni problem obstoja ▫$k$▫-razsežne degenerirane koordinatizacije z enotsko razdaljo za dani graf NP-polna problema. V zadnjem delu disertacije predstavimo hevristiko za "risanje" grafov z enotsko razdaljo, ki temelji na algoritmu SPE, ki ga je leta 2003 predstavil D. K. Agrafiotis. Definiramo dilacijski koeficient in predstavimo teoretično dobljene meje zanj. Teoretične rezultate primerjamo z rezultati dobljenimi z algoritmom za risanje grafov, ki temelji na simulaciji fizikalnega modela z vzmetmi in z algoritmom, ki s pomočjo lokalne optimizacije minimizira dilacijski koeficient. Sedmo poglavje povzema rezultate objavljene v članku [B. Horvat, T. Pisanski, A. Žitnik: The dilation coefficient of a complete graph, Croat. Chem. Acta, (sprejeto), 2009].

Keywords

teorija grafov;graf z enotsko razdaljo;predstavitev;realizacija;koordinatizacija;degenerirana predstavitev;grafovski produkti;Heawoodov graf;Petersenov graf;posplošeni Petersenovi grafi;▫$I$▫-grafi;NP poln problem;dilacijski koeficient;algoritem;izomorfizem grafov;

Data

Language:	Slovenian
Year of publishing:	2009
Typology:	2.08 - Doctoral Dissertation
Organization:	UL FRI - Faculty of Computer and Information Science
Publisher:	[B. Horvat]
UDC:	519.17(043.3)
COBISS:	15190873
Views:	112
Downloads:	8
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Representations of unit-distance graphs
Secondary abstract:	The doctoral thesis describes problems concerning graphs that can be represented in the Euclidean plane (or ▫$k$▫-space) in such a way, that vertices are represented as points in the plane (▫$k$▫-space) and edges as line segments of unit lengths. Problems are observed from a computational and a mathematical point of view. In the first part of the thesis the (already known, mainly mathematical) theory of unit-distance graph representations is presented; at the same time the terminology of the results is unified and several propositions are proved. First computer aided attempts to generate small graphs with a unit-distance representation are discussed. In the following chapter the well-known graph products of ▫$k$▫-dimensional unit-distance graphs are studied. The third chapter disproves the wrong assumption that Heawood graph is not a unit-distance graph, by providing the unit-distance coordinatization of it. In the fourth chapter all degenerate unit-distance representations of the Petersen graph in the Euclidean plane are presented and some relationships among them are observed. In the following chapter generalized Petersen graphs and ▫$I$▫-graphs are observed. Necessary and sufficient conditions for two ▫$I$▫-graphs to be isomorphic are given. As a corollary it is shown that a large subclass of ▫$I$▫-graphs can be drawn with unit-distances in the Euclidean plane by using the representation with a rotational symmetry. Conjectures concerning unit-distance coordinatizations and highly-degenerate unit-distance representations of ▫$I$▫-graphs are stated and verified for all ▫$I$▫-graphs up to 2000 vertices. In the sixth chapter the decision problems that ask about the existence of a degenerate ▫$k$▫-dimensional unit-distance representation or coordinatization of a given graph are shown to be NP-complete. In the last chapter of the thesis a heuristics that draws a given graph in the Euclidean plane by minimizing the quotient of the longest and the shortest edge length is presented; see SPE algorithm in [D.Agrafiotis. Stochastic proximity embedding. J. Comput. Chem., 24 (2003) 1215-122]. The dilation coefficient of a graph is introduced and theoretically obtained bounds for the dilation coefficient of a complete graph are given. The calculated upper bounds for the dilation coefficients of complete graphs are compared to the values obtained by three graph-drawing algorithms, see [B. Horvat, T. Pisanski, A. Žitnik: The dilation coefficient of a complete graph, Croat. Chem. Acta, (accepted), 2009].
Secondary keywords:	graph theory;unit-distance graph;representation;coordinatization;degenerate representation;graph product;Heawood graph;Petersen graph;generalized Petersen graphs;▫$I$▫-graph;MP-complete problem;dilation coefficient;algorithm;graph isomorphism;
File type:	application/pdf
Type (COBISS):	Dissertation
Thesis comment:	Univ. v Ljubljani, Fak. za računalništvo in informatiko
Pages:	154 str.
ID:	24252683