  We say that a graph is intrinsically non-trivial if every spatial embedding
of the graph contains a non-trivial spatial subgraph. We prove that an
intrinsically non-trivial graph is intrinsically linked, namely every spatial
embedding of the graph contains a non-splittable 2-component link. We also show
that there exists a graph such that every spatial embedding of the graph
contains either a non-splittable 3-component link or an irreducible spatial
handcuff graph whose constituent 2-component link is split.
