# Edge contraction-like graph operation

Mathematics Asked by Emre Yolcu on October 5, 2020

Given a simple graph $$G=(V,E)$$ and an edge $$uvin E$$, 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 $$v$$.

Does this graph operation have a name or has it ever come up in the literature?

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