# Is there a name for this graph-theoretic concept?

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?

• 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. – Kundor Feb 21 '16 at 20:02
• Sorry I didn't make this explicit, but I was thinking about simple graphs. – Vik78 Feb 21 '16 at 20:47