  We classify positive tight contact structures, up to isotopy fixing the
boundary, on the manifolds $N=M(D^{2}; r_1, r_2)$ with minimal convex boundary
of slope $s$ and Giroux torsion 0 along $\partial N$, where $r_1,r_2\in
(0,1)\cap\mathbb{Q}$, in the following cases:
  (1) $s\in(-\infty, 0)\cup[2, +\infty)$;
  (2) $s\in[0, 1)$ and $r_1,r_2\in [1/2,1)$;
  (3) $s\in[1, 2)$ and $r_1,r_2\in(0,1/2)$;
  (4) $s=\infty$ and $r_1=r_2=1/2$.
  We also classify positive tight contact structures, up to isotopy fixing the
boundary, on $M(D^2;1/2,1/2)$ with minimal convex boundary of arbitrary slope
and Giroux torsion greater than 0 along the boundary.
