Reconstructing hypergraph matching polynomials

TL;DR

This paper extends the graph reconstruction theorem to hypergraphs, showing that the matching polynomial of a hypergraph can be reconstructed from induced sub-hypergraphs of a specific size, generalizing Godsil's result.

Contribution

It proves a hypergraph analogue of Godsil's identity, enabling reconstruction of hypergraph matching polynomials from smaller induced sub-hypergraphs, and establishes optimality of the sub-hypergraph size.

Findings

Reconstruction of hypergraph matching polynomial from induced sub-hypergraphs.

Generalization of Godsil's graph result to hypergraphs.

Optimality of the sub-hypergraph size for reconstruction.

Abstract

By utilizing the recently developed hypergraph analogue of Godsil's identity by the second author, we prove that for all $n \geq k \geq 2$, one can reconstruct the matching polynomial of an $n$-vertex $k$-uniform hypergraph from the multiset of all induced sub-hypergraphs on $\lfloor \frac{k-1}{k}n \rfloor + 1$ vertices. This generalizes the well-known result of Godsil on graphs in 1981 to every uniform hypergraph. As a corollary, we show that for every graph $F$, one can reconstruct the number of $F$-factors in a graph under analogous conditions. We also constructed examples that imply the number $\lfloor \frac{k-1}{k}n \rfloor + 1$ is the best possible for all $n\geq k \geq 2$ with $n$ divisible by $k$.