We say that a graph is intrinsically knotted or completely 3-linked if every embedding of the graph into the 3-sphere contains a nontrivial knot or a 3-component link each of whose 2-component sublinks is nonsplittable. We show that a graph obtained from the complete graph on seven vertices by a finite sequence of Y-exchanges and Y -exchanges is a minor-minimal intrinsically knotted or completely 3-linked graph.

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It is well-known that for any link L, twice the unknotting number of L is less than or equal to the crossing number of L. Taniyama characterized the links which satisfy the equality. We characterize the links where twice the unknotting number is equal to the crossing number minus one. As a corollary, we show that for any link L with twice the unknotting number of L is greater than or equal to the crossing number of L minus two, every minimal diagram of L realizes the unknotting number.

A finite set of specific knotted double-handcuff graphs is shown to be minimal among those which produce all projections of knotted double-handcuff graphs. In addition, we show that a double-handcuff graph has no strongly almost trivial spatial embeddings.