A reconstruction problem related to balance equations II: The general case

A modified k-deck of a graph G, first introduced in (Krasikov and Roditty, 1987), is obtained by removing k edges of G in all possible ways, and adding k (not necessarily new) edges in all possible ways. Krasikov and Roditty asked if it was possible to construct the usual k-edge deck of a graph from its modified k-deck. In (Thatte, to appear), the author solved this problem for the case when k = 1. In this paper, the problem is completely solved for arbitrary k. The proof makes use of the k-edge version of Lovászʹs result and the eigenvalues of certain matrix related to the Johnson graph.