The two-state spinless Falicov-Kimball model on a one-dimensional latt
ice is studied by means of well-controlled numerical procedures. Restr
icted phase diagrams in the grand-canonical ensemble and at zero tempe
rature are constructed. The evolution of these phase diagrams, as the
interaction parameter U is varied, including the band structures corre
sponding to configurations of localized particles (ions) and densities
of mobile particles (electrons), is monitored. The changes observed e
nable us to draw conclusions for the global structure of the phase dia
gram and to make a qualitative plot of such a diagram. Particular atte
ntion is paid to the possibility of conductor-insulator transitions. T
he results obtained suggest that the system is an insulator for all U
only if the electron and ion densities rho(c), rho(i) satisfy the cond
itions: rho(e) + rho(i) = 1, \rho(i) - 1/2\ less-than-or-equal-to delt
a < 1/2. For all other pairs (rho(c), rho(i)) there is a critical U at
which the system undergoes a conductor-insulator transition.