Squares of Low Maximum Degree

TL;DR

This paper presents efficient algorithms for the Square Root problem, showing it can be solved in linear time for graphs with maximum degree 5 and in polynomial time for degrees up to 6.

Contribution

It introduces new polynomial-time algorithms for the Square Root problem on graphs with bounded maximum degree, expanding the classes of graphs where the problem is efficiently solvable.

Findings

Square Root is solvable in O(n) time for maximum degree 5 graphs.

Square Root is solvable in O(n^4) time for maximum degree 6 graphs.

The complexity of the problem is reduced for graphs with small maximum degree.

Abstract

A graph H is a square root of a graph G if G can be obtained from H by adding an edge between any two vertices in H that are of distance 2. The Square Root problem is that of deciding whether a given graph admits a square root. This problem is only known to be NP-complete for chordal graphs and polynomial-time solvable for non-trivial minor-closed graph classes and a very limited number of other graph classes. We prove that Square Root is O(n)-time solvable for graphs of maximum degree 5 and O(n^4)-time solvable for graphs of maximum degree at most 6.