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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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