  The Euclidean Steiner Minimal Tree problem takes as input a set $\mathcal P$
of points in the Euclidean plane and finds the minimum length network
interconnecting all the points of $\mathcal P$. In this paper, in continuation
to the works of Du et al. and Weng et al., we study Euclidean Steiner Minimal
Tree when $\mathcal P$ is formed by the vertices of a pair of regular,
concentric and parallel $n$-gons. We restrict our attention to the cases where
the two polygons are not very close to each other. In such cases, we show that
Euclidean Steiner Minimal Tree is polynomial-time solvable, and we describe an
explicit structure of a Euclidean Steiner minimal tree for $\mathcal P$. We
also consider point sets $\mathcal P$ of size $n$ where the number of input
points not on the convex hull of $\mathcal P$ is $f(n) \leq n$. We give an
exact algorithm with running time $2^{\mathcal{O}(f(n)\log n)}$ for such input
point sets $\mathcal P$. Note that when $f(n) = \mathcal{O}(\frac{n}{\log n})$,
our algorithm runs in single-exponential time, and when $f(n) = o(n)$ the
running time is $2^{o(n\log n)}$ which is better than the known algorithm
stated in Hwang et al. We know that no FPTAS exists for Euclidean Steiner
Minimal Tree unless P=NP, as shown by Garey et al. On the other hand FPTASes
exist for Euclidean Steiner Minimal Tree on convex point sets, as given by
Scott Provan. In this paper, we show that if the number of input points in
$\mathcal P$ not belonging to the convex hull of $\mathcal P$ is
$\mathcal{O}(\log n)$, then an FPTAS exists for Euclidean Steiner Minimal Tree.
In contrast, we show that for any $\epsilon \in (0,1]$, when there are
$\Omega(n^{\epsilon})$ points not belonging to the convex hull of the input
set, then no FPTAS can exist for Euclidean Steiner Minimal Tree unless P=NP.
