On plane graphs with link component number equal to the nullity

Authors:
Yuefeng Lin;S. D. Noble;Xian'an Jin;Wenfang Cheng
Affiliations:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, PR China and Zhangzhou City Vocational College, Zhangzhou 363000, PR China;Department of Mathematical Sciences, Brunel University, Kingston Lane, Uxbridge, UB8 3PH, UK;School of Mathematical Sciences, Xiamen University, Xiamen 361005, PR China;School of Mathematical Sciences, Xiamen University, Xiamen 361005, PR China
Venue:
Discrete Applied Mathematics
Year:
2012

Citing 3
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Abstract

In this paper, we study connected plane graphs with link component number equal to the nullity and call them near-extremal graphs. We first study near-extremal graphs with minimum degree at least 3 and prove that a connected plane graph G with minimum degree at least 3 is a near-extremal graph if and only if G is isomorphic to K"4, the complete graph with 4 vertices. The result is obtained by studying general graphs using the knowledge of bicycle space and the Tutte polynomial. Then a simple algorithm is given to judge whether a connected plane graph is a near-extremal graph or not. Finally we study the construction of near-extremal graphs and prove that all near-extremal graphs can be constructed from a loop and K"4 by two graph operations.