  In 2000, Brightwell and Winkler characterised dismantlable graphs as the
graphs $H$ for which the Hom-graph ${\rm Hom}(G,H)$, defined on the set of
homomorphisms from $G$ to $H$, is connected for all graphs $G$. This shows that
the reconfiguration version ${\rm Recon_{Hom}}(H)$ of the $H$-colouring
problem, in which one must decide for a given $G$ whether ${\rm Hom}(G,H)$ is
connected, is trivial if and only if $H$ is dismantlable.
  We prove a similar starting point for the reconfiguration version of the
$H$-extension problem. Where ${\rm Hom}(G,H;p)$ is the subgraph of the
Hom-graph ${\rm Hom}(G,H)$ induced by the $H$-colourings extending the
$H$-precolouring $p$ of $G$, the reconfiguration version ${\rm Recon_{Ext}(H)}$
of the $H$-extension problem asks, for a given $H$-precolouring $p$ of a graph
$G$, if ${\rm Hom}(G,H;p)$ is connected. We show that the graphs $H$ for which
${\rm Hom}(G,H;p)$ is connected for every choice of $(G,p)$ are exactly the
${\rm NU}$ graphs. This gives a new characterisation of ${\rm NU}$ graphs, a
nice class of graphs that is important in the algebraic approach to the ${\rm
CSP}$-dichotomy.
  We further give bounds on the diameter of ${\rm Hom}(G,H;p)$ for ${\rm NU}$
graphs $H$, and show that shortest path between two vertices of ${\rm
Hom}(G,H;p)$ can be found in parameterised polynomial time. We apply our
results to the problem of shortest path reconfiguration, significantly
extending recent results.
