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

Let GG be a connected plane graph, D(G)D(G) be the corresponding link diagram via medial construction, and μ(D(G))μ(D(G)) be the number of components of the link diagram D(G)D(G). In this paper, we first provide an elementary proof that μ(D(G))≤n(G)+1μ(D(G))≤n(G)+1, where n(G)n(G) is the nullity of GG. Then we lay emphasis on the extremal graphs, i.e. the graphs with μ(D(G))=n(G)+1μ(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 K1K1 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.