Characterizations of p-groups whose power graphs satisfy certain connectivity conditions

Let [Formula: see text] be an undirected and simple graph. A set [Formula: see text] of vertices in [Formula: see text] is called a cyclic vertex cutset of [Formula: see text] if [Formula: see text] is disconnected and has at least two components containing cycles. If [Formula: see text] has a cyclic vertex cutset, then it is said to be cyclically separable. The cyclic vertex connectivity of [Formula: see text] is the minimum of cardinalities of the cyclic vertex cutsets of [Formula: see text]. The power graph [Formula: see text] of a group [Formula: see text] is the undirected and simple graph whose vertices are the elements [Formula: see text] and two vertices are adjacent if one of them is the power of other in [Formula: see text]. In this paper, we first characterize the finite [Formula: see text]-groups ([Formula: see text] is a prime number) whose power graphs are cyclically separable in terms of their maximal cyclic subgroups. Then, we characterize the finite [Formula: see text]-groups whose power graphs have equal vertex connectivity and cyclic vertex connectivity.