  We study the Steiner Tree problem on unit disk graphs. Given a $n$ vertex
unit disk graph $G$, a subset $R\subseteq V(G)$ of $t$ vertices and a positive
integer $k$, the objective is to decide if there exists a tree $T$ in $G$ that
spans over all vertices of $R$ and uses at most $k$ vertices from $V\setminus
R$. The vertices of $R$ are referred to as terminals and the vertices of
$V(G)\setminus R$ as Steiner vertices. First, we show that the problem is
NP-Hard. Next, we prove that the Steiner Tree problem on unit disk graphs can
be solved in $n^{O(\sqrt{t+k})}$ time. We also show that the Steiner Tree
problem on unit disk graphs parameterized by $k$ has an FPT algorithm with
running time $2^{O(k)}n^{O(1)}$. In fact, the algorithms are designed for a
more general class of graphs, called clique-grid graphs. We mention that the
algorithmic results can be made to work for the Steiner Tree on disk graphs
with bounded aspect ratio. Finally, we prove that the Steiner Tree on disk
graphs parameterized by $k$ is W[1]-hard.
