  Let $G$ be a graph, and $v\in V(G)$ and $S\subseteq V(G)\backslash v$ of size
at least $k$. An important result on graph connectivity due to Perfect states
that, if $v$ and $S$ are $k$-linked, then a $(k-1)$-link between a vertex $v$
and $S$ can be extended to a $k$-link between $v$ and $S$ such that the
endvertices of the $(k-1)$-link are also the endvertices of the $k$-link. We
begin by proving a generalization of Perfect's result by showing that, if two
disjoint sets $S_1$ and $S_2$ are $k$-linked, then a $t$-link ($t< k$) between
two disjoint sets $S_1$ and $S_2$ can be extended to a $k$-link between $S_1$
and $S_2$ such that the endvertices of the $t$-link are preserved in the
$k$-link.
  Next, we are able to use these results to show that a 3-connected claw-free
graph always has a cycle passing through any given five vertices but avoiding
any other one specified vertex. We also show that this result is sharp by
exhibiting an infinite family of 3-connected claw-free graphs in which there is
no cycle containing a certain set of six vertices but avoiding a seventh
specified vertex. A direct corollary of our main result shows that, a
3-connected claw-free graph has a topological wheel minor $W_k$ with $k\le 5$
if and only if it has a vertex of degree at least $k$.
  Finally, we also show that a graph polyhedrally embedded in a surface always
has a cycle passing through any given three vertices but avoiding any other
specified vertex. The result is best possible in the sense that the polyhedral
embedding assumption is necessary, and there are infinitely many graphs
polyhedrally embedded in any surface having no cycle containing a certain set
of four vertices but avoiding a fifth specified vertex.
