  Call a colouring of a graph \emph{distinguishing} if the only automorphism of
this graph which preserves said colouring is the identity. We say that a graph
$G$ is \emph{without} $H$ if $G$ does not contain an induced subgraph
isomorphic to $H$. It was shown in \cite{gorzkow} that every locally finite
connected graph without $K_{1,n}$ admits a distinguishing edge colouring with
at most $n-1$ colours. We extend this result to locally finite connected graphs
without $K_{1,n}$.
