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
Recognizing Powers of Proper Interval, Split, and Chordal Graphs
SIAM Journal on Discrete Mathematics
Structure and linear-time recognition of 4-leaf powers
ACM Transactions on Algorithms (TALG)
Closest 4-leaf power is fixed-parameter tractable
Discrete Applied Mathematics
SIAM Journal on Discrete Mathematics
Computing square roots of trivially perfect and threshold graphs
Discrete Applied Mathematics
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Graph H is a root of graph G if there exists a positive integer k such that x and y are adjacent in G if and only if their distance in H is at most k. Motwani and Sudan [1994] proved the NP-completeness of graph square recognition and conjectured that it is also NP-complete to recognize squares of bipartite graphs. The main result of this article is to show that squares of bipartite graphs can be recognized in polynomial time. In fact, we give a polynomial-time algorithm to count the number of different bipartite square roots of a graph, although this number could be exponential in the size of the input graph. By using the ideas 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 are unique up to isomorphism. Finally, we prove the NP-completeness of recognizing cubes of bipartite graphs.