Elementarni benzenoidni grafi in nad njimi definirani grafi : doktorska disertacija

slovenian

doktorska dela

2008

Univerza v Mariboru, Fakulteta za naravoslovje in matematiko

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.

08.08.2023