Treffer: Cyclic edge and cyclic vertex connectivity of [formula omitted]-fullerene graphs.

Cyclic vertex connectivity c κ and cyclic edge connectivity c λ are two important kinds of conditional connectivity, which reflect the number of vertices or edges that can be removed before the graph is disconnected and at least two components contain a cycle, respectively. They have important applications in various networks such as computer networks or biochemical networks. In addition, a fullerene is a special kind of molecule in chemistry. A classic fullerene graph is a 3-connected cubic planar graph with only pentagonal and hexagonal faces. (4, 5, 6)-fullerene graphs are atypical fullerene graphs which also contain 4-faces. In this paper, we prove that c κ = c λ for (4 , 5 , 6) -fullerene graphs except for four exceptional graphs with order less than 16. We also give O (ν) -algorithms to determine the cyclic vertex connectivity and the cyclic edge connectivity of (4 , 5 , 6) -fullerene graphs. [ABSTRACT FROM AUTHOR]