# On Collapsing Prefix Normal Words

## TL;DR

This paper explores properties of prefix normal words, focusing on prefix normal palindromes and collapsing words, providing characterizations and connections to help address open problems in the field.

## Contribution

It introduces characterizations of prefix normal palindromes and collapsing words, extending the understanding of prefix normal words and proposing ways to decompose related open problems.

## Key findings

- Characterizations of prefix normal palindromes
- Characterizations of collapsing words
- Connection between palindromes and collapsing words

## Abstract

Prefix normal words are binary words in which each prefix has at least the same number of $\so$s as any factor of the same length. Firstly introduced by Fici and Lipt\'ak in 2011, the problem of determining the index of the prefix equivalence relation is still open. In this paper, we investigate two aspects of the problem, namely prefix normal palindromes and so-called collapsing words (extending the notion of critical words). We prove characterizations for both the palindromes and the collapsing words and show their connection. Based on this, we show that still open problems regarding prefix normal words can be split into certain subproblems.