# Minimal Absent Words in Rooted and Unrooted Trees

## TL;DR

This paper extends the concept of minimal absent words to rooted and unrooted trees with labeled edges, providing bounds on their number and algorithms for efficient computation.

## Contribution

It introduces the theory of minimal absent words for trees, establishes bounds on their size, and develops output-sensitive algorithms for their computation.

## Key findings

- Bounds of O(nσ) for rooted trees and O(n^2σ) for unrooted trees on the size of MAW sets.
- Algorithms to compute all minimal absent words in output-sensitive time.
- Bounds are tight and algorithms are efficient for large trees.

## Abstract

We extend the theory of minimal absent words to (rooted and unrooted) trees, having edges labeled by letters from an alphabet $\Sigma$ of cardinality $\sigma$. We show that the set $\text{MAW}(T)$ of minimal absent words of a rooted (resp. unrooted) tree $T$ with $n$ nodes has cardinality $O(n\sigma)$ (resp. $O(n^{2}\sigma)$), and we show that these bounds are realized. Then, we exhibit algorithms to compute all minimal absent words in a rooted (resp. unrooted) tree in output-sensitive time $O(n+|\text{MAW}(T)|)$ (resp. $O(n^{2}+|\text{MAW}(T)|)$ assuming an integer alphabet of size polynomial in $n$.