Searching for square-complementary graphs: non-existence results and complexity of recognition

TL;DR

This paper investigates the properties and recognition complexity of square-complementary graphs, establishing non-existence results, embedding properties, and computational complexity classifications.

Contribution

It proves non-existence of certain squco graphs, shows bipartite graphs can be embedded in bipartite squco graphs, and classifies recognition as graph isomorphism complete.

Findings

No squco graphs with girth 6 exist

Every bipartite graph can be embedded in a bipartite squco graph

Recognizing squco graphs is graph isomorphism complete

Abstract

A graph is square-complementary (squco, for short) if its square and complement are isomorphic. We prove that there are no squco graphs with girth 6, that every bipartite graph is an induced subgraph of a squco bipartite graph, that the problem of recognizing squco graphs is graph isomorphism complete, and that no nontrivial squco graph is both bipartite and planar. These results resolve three of the open problems posed in Discrete Math. 327 (2014) 62-75.