2024/07/10 by Kacper Kluk, Kluk, Kacper, Hoang La +3 · 1 citation
Computer Science · #Combinatorics (math.CO) #Digital Image Processing Techniques #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Graph Theory and Algorithms #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2407.07500
openalex publication_date 2024/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a problem of reconstruction of connected graphs where the input gives all subsets of size k that induce a connected subgraph. Originally introduced by Bastide et al. (WG 2023) for triples (k=3), this problem received comprehensive attention in their work, alongside a study by Qi, who provided a complete characterization of graphs uniquely reconstructible via their connected triples, i.e. no other graphs share the same set of connected triples. Our contribution consists in output-polynomial time algorithms that enumerate every triangle-free graph (resp. every graph with bounded maximum degree) that is consistent with a specified set of connected k-sets. Notably, we prove that triangle-free graphs are uniquely reconstructible, while graphs with bounded maximum degree that are consistent with the same k-sets share a substantial common structure, differing only locally. We suspect that the problem is NP-hard in general and provide a NP-hardness proof for a variant where the connectivity is specified for only some k-sets (with k at least 4).