The paper investigates the closure of the language family defined by input-driven pushdown automata (IDPDA) under the following operations: insertion (Formula presented), deletion (Formula presented), square root (Formula presented), and the first half (Formula presented). For K and L recognized by nondeterministic IDPDA, with m and with n states, respectively, insertion requires mn+2m states, as long as K is well-nested; deletion is representable with 2n states, for well-nested K; square root requires n³-O(n²) states, for well-nested L; the well-nested subset of the first half is representable with 2^O(n2) states. Without the well-nestedness constraints, non-closure is established in each case.