Reconstructing graphs with subgraph compositions

2025-03-31

AI Summary

This paper investigates the subgraph combinatorial reconstruction problem: given a graph structure and an oracle that returns multiset (t-multiset) or sum (t-sum) labels of queried subgraphs, can vertex labels be uniquely recovered? If not, the graph is *confusable*: non-isomorphic labelings yield identical combinatorial signatures. We establish the first systematic theoretical framework for combinatorial reconstructibility of graphs. Our contributions include: (i) revealing alternating reconstructibility of path-plus-leaf twin structures depending on path length parity; (ii) constructing the smallest confusable graph (6 vertices) and tree (7 vertices), along with infinite families of large non-isomorphic graphs sharing identical t-multisets; (iii) proving exponential lower bounds for brute-force reconstruction, and designing optimal-query-complexity algorithms for paths, stars, and other classes; and (iv) showing that sum-composition reconstruction is always feasible for pseudo-twin paths.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Reconstruct labeled graphs from subgraph compositions efficiently
• Determine reconstructability vs confusability for given graph structures
• Develop methods to minimize compositions needed for reconstruction

Innovation

• Generalizes string reconstruction to labeled graphs
• Uses t-multiset and t-sum compositions
• Develops efficient reconstruction methods