Graph Searches and Their End Vertices

Authors Yixin Cao , Zhifeng Wang, Guozhen Rong, Jianxin Wang

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Yixin Cao
  • Department of Computing, Hong Kong Polytechnic University, Hong Kong, China
Zhifeng Wang
  • School of Computer Science and Engineering, Central South University, Changsha, China
Guozhen Rong
  • School of Computer Science and Engineering, Central South University, Changsha, China
Jianxin Wang
  • School of Computer Science and Engineering, Central South University, Changsha, China

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Yixin Cao, Zhifeng Wang, Guozhen Rong, and Jianxin Wang. Graph Searches and Their End Vertices. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Graph search, the process of visiting vertices in a graph in a specific order, has demonstrated magical powers in many important algorithms. But a systematic study was only initiated by Corneil et al. a decade ago, and only by then we started to realize how little we understand it. Even the apparently naïve question "which vertex can be the last visited by a graph search algorithm," known as the end vertex problem, turns out to be quite elusive. We give a full picture of all maximum cardinality searches on chordal graphs, which implies a polynomial-time algorithm for the end vertex problem of maximum cardinality search. It is complemented by a proof of NP-completeness of the same problem on weakly chordal graphs. We also show linear-time algorithms for deciding end vertices of breadth-first searches on interval graphs, and end vertices of lexicographic depth-first searches on chordal graphs. Finally, we present 2^n * n^O(1)-time algorithms for deciding the end vertices of breadth-first searches, depth-first searches, and maximum cardinality searches on general graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • maximum cardinality search
  • (lexicographic) breadth-first search
  • (lexicographic) depth-first search
  • chordal graph
  • weighted clique graph
  • end vertex


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  1. Jesse Beisegel, Carolin Denkert, Ekkehard Köhler, Matjaz Krnc, Nevena Pivac, Robert Scheffler, and Martin Strehler. On the End-Vertex Problem of Graph Searches. Discrete Mathematics & Theoretical Computer Science, 21(1), 2019. URL:
  2. Philip A. Bernstein and Nathan Goodman. Power of Natural Semijoins. SIAM Journal on Computing, 10(4):751-771, 1981. URL:
  3. Anne Berry, Jean R. S. Blair, Jean Paul Bordat, and Geneviève Simonet. Graph Extremities Defined by Search Algorithms. Algorithms, 3(2):100-124, 2010. URL:
  4. Anne Berry and Jean Paul Bordat. Separability Generalizes Dirac’s Theorem. Discrete Applied Mathematics, 84(1-3):43-53, 1998. URL:
  5. Jean R. S. Blair and Barry W. Peyton. An introduction to chordal graphs and clique trees. In J. A. George, J. R. Gilbert, and J. W.-H. Liu, editors, Graph Theory and Sparse Matrix Computation, volume 56 of IMA, pages 1-29. Springer-Verlag, 1993. Google Scholar
  6. Pierre Charbit, Michel Habib, and Antoine Mamcarz. Influence of the tie-break rule on the end-vertex problem. Discrete Mathematics & Theoretical Computer Science, 16(2):57-72, 2014. URL:
  7. Derek G. Corneil. A simple 3-sweep LBFS algorithm for the recognition of unit interval graphs. Discrete Applied Mathematics, 138(3):371-379, 2004. URL:
  8. Derek G. Corneil. Lexicographic Breadth First Search - A Survey. In Juraj Hromkovic, Manfred Nagl, and Bernhard Westfechtel, editors, Graph-Theoretic Concepts in Computer Science (WG), volume 3353 of LNCS, pages 1-19. Springer, 2004. URL:
  9. Derek G. Corneil, Barnaby Dalton, and Michel Habib. LDFS-based certifying algorithm for the minimum path cover problem on cocomparability graphs. SIAM Journal on Computing, 42(3):792-807, 2013. URL:
  10. Derek G. Corneil, Ekkehard Köhler, and Jean-Marc Lanlignel. On end-vertices of Lexicographic Breadth First Searches. Discrete Applied Mathematics, 158(5):434-443, 2010. URL:
  11. Derek G. Corneil and Richard Krueger. A Unified View of Graph Searching. SIAM Journal on Discrete Mathematics, 22(4):1259-1276, 2008. URL:
  12. Derek G. Corneil, Stephan Olariu, and Lorna Stewart. Linear Time Algorithms for Dominating Pairs in Asteroidal Triple-free Graphs. SIAM Journal on Computing, 28(4):1284-1297, 1999. A preliminary version appeared in ICALP 1995. URL:
  13. Derek G. Corneil, Stephan Olariu, and Lorna Stewart. The LBFS Structure and Recognition of Interval Graphs. SIAM Journal on Discrete Mathematics, 23(4):1905-1953, 2009. URL:
  14. Gabriel A. Dirac. On rigid circuit graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 25(1):71-76, 1961. URL:
  15. Delbert R. Fulkerson and Oliver A. Gross. Incidence matrices and interval graphs. Pacific Journal of Mathematics, 15(3):835-855, 1965. URL:
  16. Philippe Galinier, Michel Habib, and Christophe Paul. Chordal Graphs and Their Clique Graphs. In Manfred Nagl, editor, Graph-Theoretic Concepts in Computer Science (WG), volume 1017 of LNCS, pages 358-371. Springer, 1995. URL:
  17. Michel Habib, Ross M. McConnell, Christophe Paul, and Laurent Viennot. Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theoretical Computer Science, 234(1-2):59-84, 2000. URL:
  18. Michael Held and Richard M. Karp. A Dynamic Programming Approach to Sequencing Problems. Journal of the Society for Industrial and Applied Mathematics, 10(1):196-210, 1962. URL:
  19. John E. Hopcroft and Robert Endre Tarjan. Efficient Planarity Testing. Journal of the ACM, 21(4):549-568, 1974. URL:
  20. Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. A preliminary version appeared in CCC 1999. URL:
  21. Donald J. Rose, Robert Endre Tarjan, and George S. Lueker. Algorithmic Aspects of Vertex Elimination on Graphs. SIAM Journal on Computing, 5(2):266-283, 1976. A preliminary version appeared in STOC 1975. URL:
  22. Ravi Sethi. Scheduling Graphs on Two Processors. SIAM Journal on Computing, 5(1):73-82, 1976. URL:
  23. Douglas R. Shier. Some aspects of perfect elimination orderings in chordal graphs. Discrete Applied Mathematics, 7(3):325-331, 1984. URL:
  24. Klaus Simon. A New Simple Linear Algorithm to Recognize Interval Graphs. In Computational Geometry - Methods, Algorithms and Applications, International Workshop on Computational Geometry CG'91, Bern, Switzerland, March 21-22, 1991, pages 289-308, 1991. URL:
  25. Robert Endre Tarjan. Depth-First Search and Linear Graph Algorithms. SIAM Journal on Computing, 1(2):146-160, 1972. A preliminary version appeared in SWAT (FOCS) 1971. URL:
  26. Robert Endre Tarjan and Mihalis Yannakakis. Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs. SIAM Journal on Computing, 13(3):566-579, 1984. With Addendum in the same journal, 14(1):254-255, 1985. URL:
  27. Marc Tedder, Derek G. Corneil, Michel Habib, and Christophe Paul. Simpler linear-time modular decomposition via recursive factorizing permutations. In Automata, Languages and Programming (ICALP), volume 5125 of LNCS, pages 634-645. Springer-Verlag, 2008. URL: