, Volume 71, Issue 2, pp 471–495 | Cite as

Linear-Time Algorithms for Tree Root Problems

  • Maw-Shang Chang
  • Ming-Tat Ko
  • Hsueh-I LuEmail author


Let T be a tree on a set V of nodes. The p-th power T p of T is the graph on V such that any two nodes u and w of V are adjacent in T p if and only if the distance of u and w in T is at most p. Given an n-node m-edge graph G and a positive integer p, the p-th tree root problem asks for a tree T, if any, such that G=T p . Given an n-node m-edge graph G, the tree root problem asks for a positive integer p and a tree T, if any, such that G=T p . Kearney and Corneil gave the best previously known algorithms for both problems. Their algorithm for the former (respectively, latter) problem runs in O(n 3) (respectively, O(n 4)) time. In this paper, we give O(n+m)-time algorithms for both problems.


Graph power Graph root Tree power Tree root Chordal graph Maximal clique Minimal node separator 



Maw-Shang Chang thanks the Institute of Information Science of Academia Sinica of Taiwan for their hospitality and support where part of this research took place.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Computer Science and Information EngineeringHungkuang UniversityTaichung CityTaiwan
  2. 2.Institute of Information ScienceAcademia SinicaTaipeiTaiwan
  3. 3.Department of Computer Science and Information EngineeringNational Taiwan UniversityTaipeiTaiwan

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