# Edge contraction-like graph operation

Given a simple graph $$G=(V,E)$$ and an edge $$uv\in E$$, the contraction of $$uv$$ refers to the replacement of the vertices $$u$$ and $$v$$ with a new vertex $$w$$ such that the edges incident on $$w$$ correspond to the ones that were incident on either $$u$$ or $$v$$. Edge contraction comes up in the study of graph minors.
Consider a similar operation: remove the edge $$uv$$, but not the vertices, and add a new vertex $$w$$ such that the edges incident on $$w$$ correspond to the ones other than $$uv$$ that are incident on either $$u$$ or $$v$$.