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

Abstract

Cyclic vertex connectivity cκ and cyclic edge connectivity cλ 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 cκ=cλ 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.