Distances in cocomparability graphs and their powers
Discrete Applied Mathematics
Locality in distributed graph algorithms
SIAM Journal on Computing
Computing roots of graphs is hard
Discrete Applied Mathematics
Algorithms for Square Roots of Graphs
SIAM Journal on Discrete Mathematics
Journal of Algorithms
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
On graph powers for leaf-labeled trees
Journal of Algorithms
Frequency Channel Assignment on Planar Networks
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Theoretical Computer Science
Linear-Time algorithms for tree root problems
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
Error compensation in leaf root problems
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
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Graph H is a root of graph G if there exists a natural number k such that xy ∈ E(G) ↔ dH(x, y) ≤ k where dH(x, y) is the length of a shortest path in H from x to y. In such a case, H is a k-th root of G and we write G = Hk and call G the k-th power of H. Motwani and Sudan proved that it is NP-complete to recognize squares of graphs and believed it is also NP-complete to recognize squares of bipartite graphs. In this paper, we show, rather surprisingly, that squares of bipartite graphs can be recognized in polynomial time. Also, we show that counting the number of different bipartite square roots of a graph can be done in polynomial time although this number could be exponential in the size of the input graph. Furthermore, we can generate all bipartite roots of a graph G in time O(max{Δ(G) · M, r(G)}) where Δ(G) is the maximum degree of G, M is the time complexity to do matrix multiplication, and r(G) is the number of different bipartite square roots of G. By using the tools developed, we are able to give a new and simpler linear time algorithm to recognize squares of trees and a new algorithmic proof that tree square roots, when they exist, are unique up to isomorphism. Finally, we prove the NP-completeness of recognition of cubes of bipartite graphs.