On graphs determining links with maximal number of components via medial construction

  • Authors:

  • Xian'an Jin;Fengming Dong;Eng Guan Tay

  • Affiliations:

  • School of Mathematical Sciences, Xiamen University, Xiamen 361005, PR China;Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, Singapore 637616, Singapore;Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, Singapore 637616, Singapore

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2009

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Abstract

Let G be a connected plane graph, D(G) be the corresponding link diagram via medial construction, and @m(D(G)) be the number of components of the link diagram D(G). In this paper, we first provide an elementary proof that @m(D(G))@?n(G)+1, where n(G) is the nullity of G. Then we lay emphasis on the extremal graphs, i.e. the graphs with @m(D(G))=n(G)+1. An algorithm is given firstly to judge whether a graph is extremal or not, then we prove that all extremal graphs can be obtained from K"1 by applying two graph operations repeatedly. We also present a dual characterization of extremal graphs and finally we provide a simple criterion on structures of bridgeless extremal graphs.