On the Reconstruction of Graph Invariants

TL;DR

This paper investigates the reconstructibility of various graph invariants, including several polynomials, from vertex-deleted decks, contributing to the ongoing effort to prove the graph reconstruction conjecture.

Contribution

The paper demonstrates that the interlace polynomial, U-polynomial, universal edge elimination polynomial, and their colored variants are reconstructible from vertex-deleted decks.

Findings

Interlace polynomial is reconstructible.

U-polynomial is reconstructible.

Universal edge elimination polynomial is reconstructible.

Abstract

The reconstruction conjecture has remained open for simple undirected graphs since it was suggested in 1941 by Kelly and Ulam. In an attempt to prove the conjecture, many graph invariants have been shown to be reconstructible from the vertex-deleted deck, and in particular, some prominent graph polynomials. Among these are the Tutte polynomial, the chromatic polynomial and the characteristic polynomial. We show that the interlace polynomial, the U -polynomial, the universal edge elimination polynomial xi and the colored versions of the latter two are reconstructible.