A note on the extendability of an isomorphism of subgraphs of a graph to an automorphism of the graph

Let $\Gamma$ be an undirected connected locally finite graph such that its automorphism group is vertex-transitive and has finite vertex stabilizers. For a vertex $v$ of $\Gamma$ and a non-negative integer $n$, let $\langle B_\Gamma(v,n)\rangle_\Gamma$ denote the subgraph of $\Gamma$ generated by the ball $B_\Gamma(v,n)$ of radius $n$ with center $v$. We prove that there exists a non-negative integer $c$ (depending only on $\Gamma$) such that, for any vertices $x$ and $y$ of $\Gamma$ and any non-negative integer $r$, if an isomorphism of $\langle B_\Gamma(x,r)\rangle_\Gamma$ onto $\langle B_\Gamma(y,r)\rangle_\Gamma$ can be extended to an isomorphism of $\langle B_\Gamma(x,r+c)\rangle_\Gamma$ onto $\langle B_\Gamma(y,r+c)\rangle_\Gamma$, then it can also be extended to an automorphism of $\Gamma$. Furthermore, we give a “formula” for $c$. In such a form the result can also be of interest for finite graphs $\Gamma$.