Abstract
Let G be a connected plane graph, D ( G ) be the corresponding link diagram via medial construction, and μ ( D ( G ) ) be the number of components of the link diagram D ( G ) . In this paper, we first provide an elementary proof that μ ( 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 μ ( 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.
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