For each vertex, branch on putting a drone either on this vertex or
one of its neighbours. For the selected vertex, mark it and its
neighbours and continue.  We do now position drones on marked
vertices.

Note that it is important that one does not copy the graph, but do it
in-place.  Running time $O(5^k + n)$. Note that by Ramsey theory,
since $\omega(G) \leq 5$, if $n \geq 442$ then this is a yes-instance.

Indeed, we do not need to invoke Ramsey, but simply observe that we
have max degree 4, the graph needs to be how big in order for it to be
yes?  30?