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1.
1-factors and characterization of reducible faces of plane elementary bipartite graphs
Andrej Taranenko, Aleksander Vesel, 2012, original scientific article

Abstract: As a general case of molecular graphs of benzenoid hydrocarbons, we study plane bipartite graphs with Kekulé structures (1-factors). A bipartite graph ▫$G$▫ is called elementary if ▫$G$▫ is connected and every edge belongs to a 1-factor of ▫$G$▫. Some properties of the minimal and the maximal 1-factor of a plane elementary graph are given. A peripheral face ▫$f$▫ of a plane elementary graph is reducible, if the removal of the internal vertices and edges of the path that is the intersection of ▫$f$▫ and the outer cycle of ▫$G$▫ results in an elementary graph. We characterize the reducible faces of a plane elementary bipartite graph. This result generalizes the characterization of reducible faces of an elementary benzenoid graph.
Keywords: mathematics, graph theory, plane elementary bipartite graph, reducible face, benzenoid graph
Published: 31.03.2017; Views: 540; Downloads: 293
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2.
On the Fibonacci dimension of partial cubes
Aleksander Vesel, 2009

Abstract: The Fibonacci dimension fdim▫$(G)$▫ of a graph ▫$G$▫ was introduced in [S. Cabello, D. Eppstein and S. Klavžar, The Fibonacci dimension of a graph, submitted] as the smallest integer ▫$d$▫ such that $G$ admits an isometric embedding into ▫$Q_d$▫, the ▫$d$▫-dimensional Fibonacci cube. A somewhat new combinatorial characterization of the Fibonacci dimension is given, which enables more comfortable proofs of some previously known results. In the second part of the paper the Fibonacci dimension of the resonance graphs of catacondensed benzenoid systems is studied. This study is inspired by the fact, that the Fibonacci cubes are precisely the resonance graphs of a subclass of the catacondensed benzenoid systems. The main result shows that the Fibonacci dimension of the resonance graph of a catacondensed benzenoid system ▫$G$▫ depends on the inner dual of ▫$G$▫. Moreover, we show that computing the Fibonacci dimension can be done in linear time for a graph of this class.
Keywords: matematika, teorija grafov, Fibonaccijeva dimenzija, delne kocke, resonančni grafi, benzenoidni sistemi, mathematics, graph theory, Fibonacci dimension, partial cubes, resonance graphs, benzenoid systems
Published: 10.07.2015; Views: 554; Downloads: 21
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3.
Binary coding of algebraic Kekulé structures of catacondensed benzenoid graphs
Damir Vukičević, Petra Žigert Pleteršek, 2008, original scientific article

Abstract: Algebraična Kekuléjeva struktura končnega katakondenziranega benzenoidnega grafa s ▫$h$▫ šestkotniki je podana z binarno kodo dolžine ▫$h$▫. Postopek je obrnljiv in sicer lahko iz binarne kode rekonstruiramo algebraično Kekuléjevo strukturo.
Keywords: matematika, kemijska teorija grafov, benzenoidni ogljikovodiki, benzenoidni grafi, Kekuléjeve strukture, Randićeve strukture, 1-faktor, binarno kodiranje, mathematics, chemical graph theory, benzenoid hydrocarbons, benzenoid graph, Kekulé structures, algebraic Kekulé structures, Randić structures, 1-factor, binary coding
Published: 10.07.2015; Views: 670; Downloads: 78
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4.
On the role of hypercubes in the resonance graphs of benzenoid graphs
Khaled Salem, Sandi Klavžar, Ivan Gutman, 2006, short scientific article

Abstract: Resonančni graf ▫$R(B)$▫ benzenoidnega grafa ▫$B$▫ ima za točke popolna prirejanja v ▫$B$▫, dve popolni prirejanji sta sosednji, če njuna simetrična razlika tvori množico povezav nekega šestkotnika v ▫$B$▫. Družina ▫$mathscr{P}$▫ paroma disjunktnih šestkotnikov benzenoidnega grafa ▫$B$▫ je resonančna v $B$, če ▫$B -- mathscr{P}$▫ vsebuje vsaj eno popolno prirejanje, ali pa je ▫$B -- mathscr{P}$▫ prazno. Dokazano je, da obstaja surjektivna preslikava ▫$f$▫ iz množice hiperkock grafa ▫$R(B)$▫ na resonančne množice v ▫$B$▫, tako da se ▫$k$▫-dimenzionalna kocka preslika na resonančno množico moči ▫$k$▫.
Keywords: matematika, teorija grafov, benzenoidni graf, popolno prirejanje, resonančni graf, hiperkocka, mathematics, graph theory, benzenoid graph, perfect matching, resonance graph, hypercube
Published: 10.07.2015; Views: 862; Downloads: 59
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6.
Characterization of reducible hexagons and fast decomposition of elementary benzenoid graphs
Andrej Taranenko, Aleksander Vesel, 2008, original scientific article

Abstract: A benzenoid graph is a finite connected plane graph with no cut vertices in which every interior region is bounded by a regular hexagon of a side length one. A benzenoid graph G is elementary if every edge belongs to a 1-factor of G. A hexagon h of an elementary benzenoid graph is reducible, if the removal of boundary edges and vertices of h results in an elementary benzenoid graph. We characterize the reducible hexagons of an elementary benzenoid graph. The characterization is the basis for an algorithm which finds the sequence of reducible hexagons that decompose a graph of this class in ▫$O(n^2)$▫ time. Moreover, we present an algorithm which decomposes an elementary benzenoid graph with at most one pericondensed component in linear time.
Keywords: mathematics, graph theory, benzenoid graphs, 1-factor, hexagons, reducible hexagons, reducible face decomposition
Published: 07.06.2012; Views: 1301; Downloads: 73
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