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Let $G$ and $H$ be graphs with vertex sets $V$ and $W$, and $f\colon V \to W$ a function. We say that $f$ preserves $k$-neighborhoods if all points that are at distance $k$ from each other in $G$ are mapped to points at distance $k$ from each other in $H$. If $f$ preserves 1-neighborhoods, it is a homomorphism, and vice-versa. Is there a name for this?

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  • $\begingroup$ The "vice-versa" needn't hold in general: a homomorphism could map an edge to a loop, in which case points at distance 1 are mapped to points at distance 0. $\endgroup$ – Kundor Feb 21 '16 at 20:02
  • $\begingroup$ Sorry I didn't make this explicit, but I was thinking about simple graphs. $\endgroup$ – Vik78 Feb 21 '16 at 20:47

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