We generalize the problem of reconstructing strings from their substring compositions first introduced by Acharya et al. in 2015 motivated by polymer-based advanced data storage systems utilizing mass spectrometry. Namely, we see strings as labeled path graphs, and as such try to reconstruct labeled graphs. For a given integer t, the subgraph compositions contain either vectors of labels for each connected subgraph of order t (t-multiset-compositions) or the sum of all labels of all connected subgraphs of order t (t-sum-composition). We ask whether, given a graph of which we know the structure and an oracle whom you can query for compositions, we can reconstruct the labeling of the graph. If it is possible, then the graph is reconstructable; otherwise, it is confusable, and two labeled graphs with the same compositions are called equicomposable. We prove that reconstructing through a brute-force algorithm is wildly inefficient, before giving methods for reconstructing several graph classes using as few compositions as possible. We also give negative results, finding the smallest confusable graphs and trees, as well as families with a large number of equicomposable non-isomorphic graphs. An interesting result occurs when twinning one leaf of a path: some paths are confusable, creating a twin out of a leaf sees the graph alternating between reconstructable and confusable depending on the parity of the path, and creating a false twin out of a leaf makes the graph reconstructable using only sum-compositions in all cases.

翻译：我们将Acharya等人于2015年首次提出的基于子串组合的字符串重构问题推广到图结构，该研究最初受基于聚合物和质谱技术的先进数据存储系统所启发。具体而言，我们将字符串视为带标签的路径图，从而尝试重构带标签的图。对于给定整数t，子图组合包含两种形式：各阶连通子图的标签向量（t-多重集组合）或所有阶连通子图的标签总和（t-和组合）。我们探讨的问题是：在已知图结构且存在可查询组合信息的神谕机条件下，能否重构图的标签配置。若可实现，则称该图可重构；否则称该图可混淆，而具有相同组合的两个带标签图被称为等组合图。我们首先证明暴力搜索算法效率极低，随后提出多种使用最少组合数重构特定图类的方法。同时给出否定性结论，包括找出最小可混淆图与树，以及具有大量非同构等组合图的图族。一个有趣的现象出现在路径图中单叶节点的孪生化过程中：部分路径图会变得可混淆；对叶节点创建真孪生时，图的可重构性会随路径奇偶性交替变化；而对叶节点创建假孪生时，该图在所有情况下仅需和组合即可实现重构。