On strongly almost trivial embeddings of graphs (Q2888880)

Recall that a spatial embedding \(f\) of a planar graph \(G\) is \textit{strongly almost trivial} (SAT) if \(f\) is nontrivial (not equivalent to a planar embedding), and there exists a projection of the embedding such that any embedding obtained from that projection, restricted to any proper subgraph of \(G\), is trivial. Previously, a few classes of graphs have been shown to be SAT. Here, the author shows that a class of graphs related to forests has all members SAT. He also shows that a connected graph \(G\), with exactly one cut edge \(e\) such that \(G\) is not homeomorphic to a handcuff graph and each connected component of \(G-e\) has at least one cycle, has no SAT embedding. Finally, he shows that the property of a graph having a strongly almost trivial embedding and the property of a graph having no strongly almost trivial embedding are both not inherited by minors.