  In this paper, we concisely represent the general solution for the TM$_{1,2}$
mixing in the type-I minimal seesaw mechanism and show phenomenological
constraints of the Yukawa matrix of neutrinos $Y$. First, conditions of $Z_{2}$
symmetry associated with TM$_{1,2}$ restrict two complex parameters. In
addition, three complex parameters are constrained from five phenomenological
parameters (two neutrino masses $m_{i}$, the mixing angle $\theta_{13}$, the
Dirac phase $\delta$, and the Majorana phase $\alpha$) and the overall phase.
As a result, there is only one free complex parameter for $Y$, except for the
right-handed neutrino masses $M_{1,2}$.
  For the TM$_{1}$ mixing with normal hierarchy, the symmetry condition is
imposed only on $Y$, and the physical parameters depend on $M_{R}$. In other
situations, the dimensionless parameters $(\theta_{13}, \delta$ and $\alpha)$
do not depend on $M_{R}$ but only on certain combinations of the components of
$Y$.
