Elementarni benzenoidni grafi in nad njimi definirani grafi : doktorska disertacija

Taranenko, Andrej

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Title:	Elementarni benzenoidni grafi in nad njimi definirani grafi : doktorska disertacija
Authors:	ID Taranenko, Andrej (Author) ID Vesel, Aleksander (Mentor) More about this mentor...
Files:	DR_Taranenko_Andrej_i2008.pdf (705,16 KB) MD5: D0CF794A6A79CC2EE206B24FC9C0A17F PID: 20.500.12556/dkum/0d35e121-bdfb-4d99-99b5-e5afa5e4fca8
Language:	Slovenian
Work type:	Dissertation
Organization:	FNM - Faculty of Natural Sciences and Mathematics
Abstract:	V disertaciji najprej proučujemo elementarne benzenoidne grafe ter se posvetimo dekompoziciji reducibilnih lic, s pomočjo katere lahko konstruiramo poljuben elementarni benzenoidni graf. Najprej s pomočjo l-faktorjev karakteriziramo reducibilna lica elementarnih benzenoidnih grafov. To so tista lica grafa, ki po odstranitvi iz grafa ohranjajo lastnost elementarnosti. S pomočjo karakterizacije reducibilnih lic podamo algoritem, ki nam za dani elementarni benzenoidni graf poišče tako zaporedje reducibilnih lic, da z njihovo odstranitvijo dobimo en sam šestkotnik. Podani algoritem je mogoče izvesti v kvadratnem času. Za družino elementarnih benzenoidnih grafov z natanko eno perikondenzirano komponento podamo linearni algoritem, ki zanje poišče dekompozicij o reducibilnih lic. Nadalje s pomočjo 4-tlakovanj podamo novo karakterizacijo elementarnih benzenoidnih grafov, saj pokažemo, da je benzenoidni graf elementaren natanko tedaj, ko zanj obstaja 4-tlakovanje. Nato se posvetimo strukturi resonančnih grafov elementarnih benzenoidnih grafov, ki ne vsebujejo koronena kot podgraf, ter zanje pokažemo dekompozicijski izrek. Še več, pokažemo, da so njihovi $\tau$ -grafi tesno povezani s 4-tlakovanji pripadajočega benzenoidnega grafa. Na koncu se posvetimo Fibonaccijevim kockam, ki so resonančni grafi fibonaccenov, ki so posebna družina katakondenziranih benzenoidnih grafov. S proučevanjem strukture Fibonaccijevih kock podamo novo karakterizacijo teh grafov. Ta karakterizacija služi kot osnova za algoritem, ki v času $0(m log n)$ prepozna, ali je dani graf Fibonaccijeva kocka. Podani algoritem te grafe prepoznava hitreje od do sedaj znanih algoritmov.
Keywords:	matematika, teorija grafov, benzenoidni graf, reducibilno lice, resonančni graf, notranji dual, hiperkocka, Fibonaccijeva kocka, disertacije
Place of publishing:	[S. l.
Publisher:	A. Taranenko]
Year of publishing:	2008
PID:	20.500.12556/DKUM-9574
UDC:	519.17(043.3)
COBISS.SI-ID:	16568328
NUK URN:	URN:SI:UM:DK:UBYYWXGN
Publication date in DKUM:	21.01.2009
Views:	7056
Downloads:	571
Metadata:
Categories:	FNM
:	TARANENKO, Andrej, 2008, Elementarni benzenoidni grafi in nad njimi definirani grafi : doktorska disertacija [online]. Doctoral dissertation. S. l. : A. Taranenko. [Accessed 14 April 2025]. Retrieved from: https://dk.um.si/IzpisGradiva.php?lang=eng&id=9574 Copy citation

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Secondary language

Language:	English
Title:	Elementary benzenoid graphs and graphs defined on them
Abstract:	In this thesis we first study elementary benzenoid graphs and focus on the reducible face decomposition which can be used to construct any elementary benzenoid graph. Using 1-factors we characterize reducible faces of elementary benzenoid graphs, i. e. faces which after their removal from the original graph preserve the property of the graph being elementary. Given this characterization we present an algorithm that finds a reducible face decomposition of the given graph in quadratic time. Moreover, for elementary benzenoid graphs with at most one pericondensed component we present a linear algorithm for finding a reducible face decomposition. With the concept of 4-tiling we give a new characterization of elementary benzenoid graphs, namely we show that a benzenoid graph is elementary it and only if it allows a 4-tiling. Next we focus on the structure of resonance graphs of elementary benzenoid graphs which do not possess a coronene as a subgraph. The structure is presented in the form of a decomposition theorem. We conclude that there is a connection between 4-tilings of elementary benzenoid graphs without a coronene as a subgraph and $\tau$ -graphs of their resonance graphs. Finally, we focus on the structure of Fibonacci cubes, which are resonance graphs of fibonaccenes - a family of catacondensed benzeniod graphs. Studying their structure enables us to give a new characterization of Fibonacci cubes which serves as a basis for a recognition algorithm of Fibonacci cubes. This algorithm with the time complexity of $0(m log n)$ is faster than any previously known algorithm.

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