The modified Toda (mToda) hierarchy is a two-component generalization of the 1-st modified KP (mKP) hierarchy, which connects the Toda hierarchy via Miura links and has two tau functions. Based on the fact that the mToda and 1-st mKP hierarchies share the same fermionic form, we firstly construct the reduction of the mToda hierarchy \(L_1(n)^M=L_2(n)^N+\sum _{l\in \mathbb {Z}}\sum _{i=1}^{m}q_{i,n}\Lambda ^lr_{i,n+1}\Delta \) and \((L_1(n)^M+L_2(n)^N)(1)=0\) , called the generalized bigraded modified Toda hierarchy, which can be viewed as a new two-component generalization of the constrained mKP hierarchy \(\mathfrak {L}^k=(\mathfrak {L}^k)_{\ge 1}+\sum _{i=1}^m \mathfrak {q}_i\partial ^{-1}\mathfrak {r}_i\partial \) . Next the relation with the Toda reduction \(\mathcal {L}_1(n)^M=\mathcal {L}_2(n)^{N}+\sum _{l\in \mathbb {Z}}\sum _{i=1}^{m}\tilde{q}_{i,n}\Lambda ^l\tilde{r}_{i,n}\) is discussed. Finally we give equivalent formulations of the Toda and mToda reductions in terms of tau functions.
