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

A modified k-deck of a graph G 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. Earlier I 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 Lovász’s result and the eigenvalues of certain matrix related to the Johnson graph. This version differs from the published version. Lemma 2.3 in the published version had a typo in one equation. Also, a long manipulation of some combinatorial expressions was skipped in the original proof of Lemma 2.3, which made it difficult to follow the proof. Here a clearer proof is given.