Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Graph Reconstruction with Connectivity Queries

Kacper Kluk, Hoang La, Marta Piecyk

TL;DR

This work investigates reconstructing labeled graphs from connectivity queries on $k$-sets. It introduces a polynomial-time enumeration framework for triangle-free graphs and a skeleton-based approach for graphs with bounded maximum degree, yielding which graphs can be reconstructed from complete sets of connected/disconnected $k$-sets. The authors prove a strong uniqueness result for large triangle-free graphs and provide NP-hardness for a partial-information variant, delineating the tractability frontier between structured classes and general instances. Overall, the paper contributes constructive algorithms, structural insights, and complexity barriers that illuminate the limits of graph reconstruction under connectivity queries.

Abstract

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).

Graph Reconstruction with Connectivity Queries

TL;DR

This work investigates reconstructing labeled graphs from connectivity queries on -sets. It introduces a polynomial-time enumeration framework for triangle-free graphs and a skeleton-based approach for graphs with bounded maximum degree, yielding which graphs can be reconstructed from complete sets of connected/disconnected -sets. The authors prove a strong uniqueness result for large triangle-free graphs and provide NP-hardness for a partial-information variant, delineating the tractability frontier between structured classes and general instances. Overall, the paper contributes constructive algorithms, structural insights, and complexity barriers that illuminate the limits of graph reconstruction under connectivity queries.

Abstract

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 (), 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 -sets. Notably, we prove that triangle-free graphs are uniquely reconstructible, while graphs with bounded maximum degree that are consistent with the same -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 -sets (with at least 4).
Paper Structure (8 sections, 19 theorems, 2 figures)

This paper contains 8 sections, 19 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Enumerating consistent graphs
  4. Triangle-free graphs
  5. Graphs with bounded maximum degree
  6. Uniqueness
  7. Uniqueness for triangle-free graphs
  8. NP-hardness

Key Result

theorem 1

Let $k$ be an integer with $k \geqslant 2$. For a complete set of connected and disconnected $k$-sets on $V$, we can enumerate every connected triangle-free graph on $V$ consistent with such sets in polynomial time in $|V|$.

Figures (2)

  • Figure 1: The auxiliary partial graph $H$. Orange edges denote the unknown edges. We ommited here $k-1$ private neighbors of each vertex in the figure. Furthermore, the vertices from $\{x_i,y_j \ | \ i,j\in[n]\}$ induce a clique with the matching $\{x_iy_i \ | \ i\in [n]\}$ removed.
  • Figure 2: The auxiliary partial graph $H$ on vertices $v,v^j,u_i,u_i^j$ (left) and on $v,v^j,w_i,w_i^j$ (right).

Theorems & Definitions (43)

  • theorem 1
  • theorem 2
  • theorem 3
  • lemma 1
  • proof
  • lemma 2: Layering
  • proof
  • proof
  • proof
  • proof
  • ...and 33 more