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Short cycle structures for graphs on surfaces and an open problem of Mohar and Thomassen

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Abstract

In this paper we investigate cycle base structures of a (weighted) graph and show that much information of short cycles is contained in an MCB (i.e., minimum cycle base). After setting up a Hall type theorem for base-transformation, we give a sufficient and necessary condition for a cycle base to be an MCB. Furthermore, we show that the structure of MCB in a (weighted) graph is unique. The property is also true for those having a longest length (although much work has been down in evaluating MCB, little is known for those having a longest length). We use those methods to find out some unknown properties for short cycles sharing particular properties in (unweighted) graphs. As applications, we determine the structures of short cycles in an embedded graph and show that there exist polynomially bounded algorithms in finding a shortest contractible cycle and a shortest two-sided cycle provided such cycles exist. Those answer an open problem of B. Mohar and C. Thomassen.

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Authors and Affiliations

  1. Department of Mathematics, East China Normal University, Shanghai, 200062, China

    Han Ren & Mo Deng

  2. College of Mathematics & Information Science, Northwest Normal University, Lanzhou, 730070, China

    Mo Deng

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  1. Han Ren
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  2. Mo Deng
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Correspondence to Han Ren.

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Han, R., Mo, D. Short cycle structures for graphs on surfaces and an open problem of Mohar and Thomassen. SCI CHINA SER A 49, 212–224 (2006). https://doi.org/10.1007/s11425-005-0012-6

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  • DOI: https://doi.org/10.1007/s11425-005-0012-6

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