Some aspects of k-apex non-commuting graph of a finite group

Let Γ be a graph and k a positive integer. Th e graph Γ is k-apex, if there exist (at most) k vertices v1;v2; ... ;vk of the graph Γ such that the induced graph Γ-{v1; ... ;vk} is planar. In this paper, all the groups whose non-commuting graphs are k-apex are specified, for k ≤ 5. A graph is called 1-planar if there exists its drawing in the plane such that each edge is crossed at most once. Moreover, the 1-planarity of the non-commuting graph, is discussed for AC-groups.