ArticleOriginal scientific text
Title
1-factors and characterization of reducible faces of plane elementary bipartite graphs
Authors 1, 1
Affiliations
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
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
plane elementary bipartite graph, reducible face, perfect matching, 1-factor, benzenoid graph
Bibliography
- S.J. Cyvin and I. Gutman, Kekulé Structures in Benzenoid Hydrocarbons (Springer Verlag, Heidelberg, 1988).
- A.A. Dobrynin, I. Gutman, S. Klavžar and P. Žigert, Wiener index of hexagonal systems, Acta Appl. Math. 72 (2002) 247-294, doi: 10.1023/A:1016290123303.
- I. Gutman and S.J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons (Springer Verlag, Berlin, 1989).
- P. Hansen and M. Zheng, A linear algorithm for perfect matching in hexagonal systems, Discrete Math. 122 (2002) 179-196, doi: 10.1016/0012-365X(93)90294-4.
- S. Klavžar and M. Kovše, The Lattice dimension of benzenoid systems, MATCH Commun. Math. Comput. Chem. 56 (2006) 637-648.
- S. Klavžar, A. Vesel, P. Žigert and I. Gutman, Binary coding of Kekulé structures of catacondensed benzenoid hydrocarbons, Comput. & Chem. 25 (2001) 569-575, doi: 10.1016/S0097-8485(01)00068-7.
- S. Klavžar, A. Vesel and P. Žigert, On resonance graphs of catacondensed hexagonal graphs: structure, coding, and hamilton path algorithm, MATCH Commun. Math. Comput. Chem. 49 (2003) 100-116.
- P.C.B. Lam and H. Zhang, A distributive lattice on the set of perfect matchings of a plane biparite graph, Order 20 (2003) 13-29, doi: 10.1023/A:1024483217354.
- L. Lovász and M.D. Plummer, Matching Theory (North-Holland, 1986).
- I. Pesek and A. Vesel, Visualization of the resonance graphs of benzenoid graphs, MATCH Commun. Math. Comput. Chem. 58 (2007) 215-232.
- K. Salem and S. Klavžar, On plane bipartite graphs without fixed edges, Appl. Math. Lett. 20 (2007) 813-816, doi: 10.1016/j.aml.2006.08.014.
- A. Taranenko and A. Vesel, Characterization of reducible hexagons and fast decomposition of elementary benzenoid graphs, Discrete Appl. Math. 156 (2008) 1711-1724, doi: 10.1016/j.dam.2007.08.029.
- A. Taranenko and A. Vesel, On Elementary Benzenoid Graphs: New Characterization and Structure of Their Resonance Graphs, MATCH Commun. Math. Comput. Chem. 60 (2008) 193-216.
- F. Zhang, X. Guo and R. Chen, Z-transformation graphs of perfect matchings of hexagonal systems, Discrete Math. 72 (1988) 405-415, doi: 10.1016/0012-365X(88)90233-6.
- H. Zhang, P.C.B. Lam and W.C. Shiu, Resonance graphs and a binary coding for the 1-factors of benzenoid systems, SIAM J. Discret. Math. 22 (2008) 971-984, doi: 10.1137/070699287.
- H. Zhang and F. Zhang, The rotation graphs of perfect matchings of plane bipartite graphs, Discrete Appl. Math. 73 (1997) 5-12, doi: 10.1016/S0166-218X(96)00024-8.
- H. Zhang and F. Zhang, Plane elementary bipartite graphs, Discrete Appl. Math. 105 (2000) 291-311, doi: 10.1016/S0166-218X(00)00204-3.
Pages:
289-297
Main language of publication
English
Received
2010-11-30
Accepted
2011-05-11
Published
2012
DOI: 10.7151/dmgt.1607
Exact and natural sciences