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

The constrained KP hierarchy and the bigraded Toda hierarchy of $(M,1)$-type