{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,4]],"date-time":"2026-10-04T14:31:49Z","timestamp":1791124309840,"version":"4.1.0"},"reference-count":11,"publisher":"Association for Computing Machinery (ACM)","issue":"2","content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Algorithms"],"published-print":{"date-parts":[[2006,4]]},"abstract":"<jats:p>\n                    Graph\n                    <jats:italic>H<\/jats:italic>\n                    is a root of graph\n                    <jats:italic>G<\/jats:italic>\n                    if there exists a positive integer\n                    <jats:italic>k<\/jats:italic>\n                    such that\n                    <jats:italic>x<\/jats:italic>\n                    and\n                    <jats:italic>y<\/jats:italic>\n                    are adjacent in\n                    <jats:italic>G<\/jats:italic>\n                    if and only if their distance in\n                    <jats:italic>H<\/jats:italic>\n                    is at most\n                    <jats:italic>k<\/jats:italic>\n                    . 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.\n                  <\/jats:p>","DOI":"10.1145\/1150334.1150337","type":"journal-article","created":{"date-parts":[[2006,10,18]],"date-time":"2006-10-18T14:11:32Z","timestamp":1161180692000},"page":"178-208","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":30,"title":["Bipartite roots of graphs"],"prefix":"10.1145","volume":"2","author":[{"given":"Lap Chi","family":"Lau","sequence":"first","affiliation":[{"name":"University of Toronto, Toronto, Ont., Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2006,4]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1016\/0095-8956(74)90074-4","article-title":"Characterization of n-path graphs and of graphs having nth root","volume":"16","author":"Escalante F.","year":"1974","unstructured":"Escalante , F. , Montejano , L. , and Rojano , T. 1974 . Characterization of n-path graphs and of graphs having nth root . J. Combin. Theory B 16 , 282 -- 289 . Escalante, F., Montejano, L., and Rojano, T. 1974. Characterization of n-path graphs and of graphs having nth root. J. Combin. Theory B 16, 282--289.","journal-title":"J. Combin. Theory B"},{"key":"e_1_2_1_2_1","unstructured":"Garey M. R. and Johnson D. S. 1979. Computers and Intractability---A Guide to the Theory of NP-Completeness. Freeman Oxford UK.   Garey M. R. and Johnson D. S. 1979. Computers and Intractability---A Guide to the Theory of NP-Completeness. Freeman Oxford UK."},{"key":"e_1_2_1_3_1","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1016\/S0021-9800(68)80079-1","article-title":"The square root of a digraph","volume":"5","author":"Geller D. P.","year":"1968","unstructured":"Geller , D. P. 1968 . The square root of a digraph . J. Combin. Theory B 5 , 320 -- 321 . Geller, D. P. 1968. The square root of a digraph. J. Combin. 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C.","year":"1960","unstructured":"Ross , I. C. , and Harary , F. 1960 . The square of a tree . Bell Syst. Tech. J. 39 , 641 -- 647 . Ross, I. C., and Harary, F. 1960. The square of a tree. Bell Syst. Tech. J. 39, 641--647.","journal-title":"Bell Syst. Tech. J."},{"key":"e_1_2_1_11_1","volume-title":"Ed. 2","author":"West D.","unstructured":"West , D. 2001. Introduction to Graph Theory , Ed. 2 , Prentice-Hall , Englewood Cliffs, NJ . West, D. 2001. 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