# Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2

## TL;DR

This paper presents polynomial-time algorithms to determine if a graph has an outerplanar square root or a square root with pathwidth at most 2, addressing a classical NP-complete problem in specific graph classes.

## Contribution

It introduces new polynomial-time algorithms for deciding the existence of outerplanar and pathwidth at most 2 square roots of a given graph.

## Key findings

- Polynomial-time algorithm for outerplanar square root recognition
- Polynomial-time algorithm for square root with pathwidth at most 2
- Advances understanding of graph root problems in restricted classes

## Abstract

Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial effort has been dedicated to deciding whether a given graph has a square root that belongs to a particular graph class. There are both polynomial-time solvable and NP-complete cases, depending on the graph class. We contribute with new results in this direction. Given an arbitrary input graph G, we give polynomial-time algorithms to decide whether G has an outerplanar square root, and whether G has a square root that is of pathwidth at most 2.