  We say that a set of pairs of disjoint cycles $\Lambda(G)$ of a graph $G$ is
linked if for any spatial embedding $f$ of $G$ there exists an element
$\lambda$ of $\Lambda(G)$ such that the $2$-component link $f(\lambda)$ is
nonsplittable, and also say minimally linked if none of its proper subsets are
linked. In this paper, (1) we show that the set of all pairs of disjoint cycles
of $G$ is minimally linked if and only if $G$ is essentially same as a graph in
the Petersen family, and (2) for any two integers $p,q\ge 3$, we exhibit a
minimally linked set of Hamiltonian $(p,q)$-pairs of cycles of the complete
graph $K_{p+q}$ with at most eighteen elements.
