Characterizing and recognizing exact-distance squares of graphs

For a graph G = ( V , E ) , its exact-distance square , G [1 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 [1 2 ] . We give a characterization of graphs having an exact-distance square root, our characterization easily leading to a polynomialtime 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. (c) 2023 Elsevier B.V. All rights reserved.