Characterizing and recognizing exact-distance squares of graphs

TL;DR

This paper characterizes graphs that are exact-distance squares of other graphs, provides recognition algorithms, and explores the complexity and uniqueness of such roots, contrasting with known results on usual graph squares.

Contribution

It offers a characterization and polynomial-time recognition algorithms for graphs with exact-distance square roots, and analyzes the complexity and non-uniqueness of these roots.

Findings

Polynomial-time recognition for graphs with exact-distance square roots

NP-completeness of recognizing bipartite exact-distance square roots

Existence of multiple non-isomorphic tree roots for some graphs

Abstract

For a graph $G=(V,E)$, its exact-distance square, $G^{[\sharp 2]}$, is the graph with vertex set $V$ and with an edge between vertices $x$ and $y$ if and only if $x$ and $y$ have distance (exactly) $2$ in $G$. The graph $G$ is an exact-distance square root of $G^{[\sharp 2]}$. We give a characterization of graphs having an exact-distance square root, our characterization easily leading to a polynomial-time recognition algorithm. We show that it is NP-complete to recognize graphs with a bipartite exact-distance square root. These two results strongly contrast known results on (usual) graph squares. We then characterize graphs having a tree as an exact-distance square root, and from this obtain a polynomial-time recognition algorithm for these graphs. Finally, we show that, unlike for usual square roots, a graph might have (arbitrarily many) non-isomorphic exact-distance square roots which are trees.