An unknotting theorem for delta and sharp edge‐homotopy

Abstract

Two spatial embeddings of a graph are said to be delta (resp. sharp) edgehomotopic if they are transformed into each other by self delta (resp. sharp) moves and ambient isotopies. We show that any two spatial embeddings of a graph are delta (resp. sharp) edge-homotopic if and only if the graph does not contain a subgraph which is homeomorphic to the theta graph or the disjoint union of two 1-spheres, or equivalently G is homeomorphic to a bouquet.