Polynomial reconstruction of the matching polynomial

TL;DR

This paper investigates whether the matching polynomial of a graph can be reconstructed from subgraphs, proving it for certain graphs and disproving it for others, thereby advancing understanding of graph polynomial reconstructibility.

Contribution

It establishes the polynomial reconstructibility of the matching polynomial for graphs with pendant edges and provides counterexamples for other graphs, clarifying the scope of reconstructibility.

Findings

Matching polynomial is reconstructible for graphs with pendant edges.

Counterexamples show non-reconstructibility for some graphs.

The paper consolidates previous results on matching polynomial reconstructibility.

Abstract

The matching polynomial of a graph is the generating function of the numbers of its matchings with respect to their cardinality. A graph polynomial is polynomial reconstructible, if its value for a graph can be determined from its values for the vertex-deleted subgraphs of the same graph. This note discusses the polynomial reconstructibility of the matching polynomial. We collect previous results, prove it for graphs with pendant edges and disprove it for some graphs.