We show that there is a series of topological string theories whose integrable structure is described by the Toda lattice hierarchy. The monodromy group of the Frobenius manifold for the matter sector is an extension of the ane Weyl group f W(A 1 N ) i n troduced by Dubrovin. These models are generalizations of the topological CP 1 string theory with scaling violation. The logarithmic Hamiltonians generate ows for the puncture operator and its descendants. We derive the string equation from the constraints on the Lax and the Orlov operators. The constraints are of dierent t ype from those for the c = 1 string theory. Higher genus expansion is obtained by considering the Lax operator in matrix form.