Polynomial time recognition of squares of ptolemaic graphs and 3-sun-free split graphs

TL;DR

This paper presents polynomial time algorithms for recognizing graphs that are squares of ptolemaic graphs and 3-sun-free split graphs, solving NP-complete problems for these special cases.

Contribution

It introduces the first polynomial time algorithms for recognizing squares of ptolemaic graphs and 3-sun-free split graphs, including minimal edge root computation.

Findings

Polynomial time algorithm for ptolemaic square root recognition

Characterization and recognition of 3-sun-free split square roots

Efficient computation of minimal edge roots

Abstract

The square of a graph $G$, denoted $G^2$, is obtained from $G$ by putting an edge between two distinct vertices whenever their distance is two. Then $G$ is called a square root of $G^2$. Deciding whether a given graph has a square root is known to be NP-complete, even if the root is required to be a chordal graph or even a split graph. We present a polynomial time algorithm that decides whether a given graph has a ptolemaic square root. If such a root exists, our algorithm computes one with a minimum number of edges. In the second part of our paper, we give a characterization of the graphs that admit a 3-sun-free split square root. This characterization yields a polynomial time algorithm to decide whether a given graph has such a root, and if so, to compute one.