Polynomial invariants for rooted trees related to their random destruction

TL;DR

This paper introduces polynomial invariants for rooted trees inspired by random destruction processes, demonstrating that some invariants uniquely identify trees while others do not.

Contribution

It presents new recursive formulas and identities for polynomial invariants, proving the completeness of certain invariants in distinguishing rooted trees.

Findings

Invariants P and S are complete and distinguish rooted trees.

Recursion formulas and identities relate the invariants.

Counterexamples show A is not complete; M's completeness remains open.

Abstract

We consider three bivariate polynomial invariants $P$, $A$, and $S$ for rooted trees, as well as a trivariate polynomial invariant $M$. These invariants are motivated by random destruction processes such as the random cutting model or site percolation on rooted trees. We exhibit recursion formulas for the invariants and identities relating $P$, $S$, and $M$. The main result states that the invariants $P$ and $S$ are complete, that is they distinguish rooted trees (in fact, even rooted forests) up to isomorphism. The proof method relies on the obtained recursion formulas and on irreducibility of the polynomials in suitable unique factorization domains. For $A$, we provide counterexamples showing that it is not complete, although that question remains open for the trivariate invariant $M$.