与平衡方程相关的重建问题-II：一般情况
A reconstruction problem related to balance equations-II: the general case
摘要：
通过以所有可能的方式移除$G$的$k$边，并且以所有可能的方式添加$k$（不一定是新的）边，来获得图$G$的修改的$k$ -甲板。Krasikov和Roditty询问是否有可能从图的修改后的$k$ -deck构造图的通常的$k$ -edge deck。早些时候我解决了这个问题的情况下，$k=1$。本文对任意$k$完全解决了这个问题。利用Lovász结果的$k$ -边形式和与Johnson图有关的矩阵的特征值，证明了这一定理.这个版本不同于出版的版本。引理2.3在已发表的版本中有一个公式打错了。此外，在引理2.3的原始证明中跳过了对一些组合表达式的长时间操作，这使得证明很难进行。这里给出了一个更清楚的证明。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
