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

• Published in: Discrete Applied Mathematics Vol. 157; no. 14; pp. 3099 - 3110
• Main Authors: Jin, Xian’an, Dong, Fengming, Tay, Eng Guan
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 28.07.2009; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Combinatorics; Combinatorics. Ordered structures; Component number; Computer science; control theory; systems; Exact sciences and technology; Extremal characterization; Information retrieval. Graph; Link diagram; Mathematics; Plane graph; Sciences and techniques of general use; Theoretical computing; Link diagram; Extremal characterization; Component number; Plane graph; Construction; Computer theory; Plane; Connected graph; Algorithm; Optimization; Diagram; Characterization; Proof; Combinatorics; Maximal graph; Link
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2009.06.006

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.