# The Bipartition Polynomial of a Graph: Reconstruction, Decomposition, and Applications

## TL;DR

This paper introduces the bipartition polynomial as a versatile graph invariant that generalizes many known polynomials and demonstrates its reconstructibility from subgraph polynomials, enhancing graph property analysis.

## Contribution

It establishes the bipartition polynomial as a unifying tool for various graph polynomials and proves its polynomial reconstructibility from edge-deleted subgraphs.

## Key findings

- Bipartition polynomial generalizes multiple graph polynomials
- It can be used to prove various graph properties
- The polynomial is reconstructible from subgraphs

## Abstract

The bipartition polynomial of a graph is a generalization of many other graph polynomials, including the domination, Ising, matching, independence, cut, and Euler polynomial. We show in this paper that it is also a powerful tool for proving graph properties. In addition, we can show that the bipartition polynomial is polynomially reconstructible, which means that we can recover it from the multiset of bipartition polynomials of one-edge-deleted subgraphs.