  In this paper, we extend the matrix-resolvent method to the study of the
Dubrovin--Zhang type tau-functions for the constrained KP hierarchy and the
bigraded Toda hierarchy of $(M,1)$-type. We show that the Dubrovin--Zhang type
tau-function of an arbitrary solution to the bigraded Toda hierarchy of
$(M,1)$-type is a Dubrovin--Zhang type tau-function for the constrained KP
hierarchy, which generalizes the result in [10, 35] for the Toda lattice
hierarchy and the NLS hierarchy corresponding to the $M=1$ case.
