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Characterizing and recognizing exact-distance squares of graphs

Yandong Bai
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Pedro Cortes
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Danile Quiroz
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For a graph G = (V, E), its exact-distance square, G [♯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 [♯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 exactdistance square roots which are trees.
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hal-03840741 , version 1 (06-11-2022)


  • HAL Id : hal-03840741 , version 1


Yandong Bai, Pedro Cortes, Reza Naserasr, Danile Quiroz. Characterizing and recognizing exact-distance squares of graphs. 2022. ⟨hal-03840741⟩
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