  The ground state and excitation spectra of the $S=1$ Heisenberg spin chain
with hole hopping are investigated by means of the numerical-diagonalization
method and by the help of the Zhang-Arovas picture.
  As for the charge sector, two phases are found. It is shown that in the
weak-hopping region, a phase separation occurs, while in the strong-hopping
region, the system is in the Tomonaga-Luttinger liquid phase. Irrespective of
the presence of the spin gap, the present phase diagram is similar to that
reported for the $S=1/2$ $t$-$J$ chain with gapless spin excitation rather than
to that for the magnetically-frustrated $t$-$J$ chain with spin gap.
  For the spin sector, the string correlation is found to develop withstanding
the hole doping except in a region, whose region is characterized by a
generalized string correlation. This remains an open question to be studied in
future.
