We say that a graph is intrinsically nontrivial if every spatial embedding of the graph contains a nontrivial spatial subgraph. We prove that an intrinsically nontrivial graph is intrinsically linked, namely every spatial embedding of the graph contains a nonsplittable –component link. We also show that there exists a graph such that every spatial embedding of the graph contains either a nonsplittable –component link or an irreducible spatial handcuff graph whose constituent –component link is split.
"An intrinsic nontriviality of graphs." Algebr. Geom. Topol. 9 (1) 351 - 364, 2009. https://doi.org/10.2140/agt.2009.9.351