  The classical Kirszbraun theorem says that all $1$-Lipschitz functions
$f:A\longrightarrow \mathbb{R}^n$, $A\subset \mathbb{R}^n$, with the Euclidean
metric have a $1$-Lipschitz extension to $\mathbb{R}^n$. For metric spaces
$X,Y$ we say that $Y$ is $X$-Kirszbraun if all $1$-Lipschitz functions
$f:A\longrightarrow Y$, $A\subset X$, have a $1$-Lipschitz extension to~$X$. We
analyze the case when $X$ and $Y$ are graphs with the usual path metric. We
prove that $\mathbb{Z}^d$-Kirszbraun graphs are exactly graphs that satisfies a
certain Helly property. We also consider complexity aspects of these
properties.
