We present the class of models of a nonmagnetic impurity in an S = 1/2 generalized ladder with an Affleck-Kennedy-Lieb-Tasaki-type (AKLT-type) valence-bond ground state, and of an S = 1/2 impurity in the S = 1 AKLT chain. The ground state in the presence of impurity can be found exactly. The recently studied phenomenon of local enhancement of antiferromagnetic correlations around the impurity is absent for this family of models.