733 KB123 KB0 KB2008Taranenko, AndrejVesel, AleksanderVIII, 83 str., 30 cmCOBISSID:16568328URN:URN:NBN:SI:doc-3U7Z5VFZslA. Taranenkovisokošolska delabenzenoidni grafiDisertacijeElementarni benzenoidni grafiFibonaccijeva kockaGrafihiperkockanotranji dualreducibilno liceresonančni grafiteorija grafovElementarni benzenoidni grafi in nad njimi definirani grafi| doktorska disertacija|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 algorithmV 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 algoritmovTEXTvisokošolska delatheses and dissertationsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza v Mariboru, Fakulteta za naravoslovje in matematiko