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Polynomial Reconstruction Problem for Hypergraphs

Joshua Cooper, Utku Okur

Abstract

We show that, in general, the characteristic polynomial of a hypergraph is not determined by its ``polynomial deck'', the multiset of characteristic polynomials of its vertex-deleted subgraphs, thus settling the ``polynomial reconstruction problem'' for hypergraphs in the negative. The proof proceeds by showing that a construction due to Kocay of an infinite family of pairs of $3$-uniform hypergraphs which are non-isomorphic but share the same hypergraph deck, in fact, have different characteristic polynomials. The question remain unresolved for ordinary graphs.

Polynomial Reconstruction Problem for Hypergraphs

Abstract

We show that, in general, the characteristic polynomial of a hypergraph is not determined by its ``polynomial deck'', the multiset of characteristic polynomials of its vertex-deleted subgraphs, thus settling the ``polynomial reconstruction problem'' for hypergraphs in the negative. The proof proceeds by showing that a construction due to Kocay of an infinite family of pairs of -uniform hypergraphs which are non-isomorphic but share the same hypergraph deck, in fact, have different characteristic polynomials. The question remain unresolved for ordinary graphs.
Paper Structure (4 sections, 17 theorems, 85 equations, 1 figure)

This paper contains 4 sections, 17 theorems, 85 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Theorem
  4. Acknowledgement

Key Result

Theorem 2.1

There exists an infinite family of pairs of hypergraphs $\{( X^n,Y^n )\}_{n\geq 3}$, of rank $3$, that are hypomorphic, but not isomorphic.

Figures (1)

  • Figure 1: Implications between reconstruction problems for ordinary graphs, where GRP stands for Graph Reconstruction Problem and PRP stands for Polynomial Reconstruction Problem.

Theorems & Definitions (57)

  • Conjecture 1.1: Kelly, 1957; Ulam, 1960
  • Definition 1
  • Theorem 2.1
  • Definition 2: Adjacency Hypermatrix
  • Definition 3
  • Definition 4: Eigenpairs of tensors
  • Remark 1
  • Definition 5
  • Lemma 2.2
  • proof
  • ...and 47 more