Skip to main content
SpringerLink
Log in

Graph Search Trees and Their Leaves

  • Conference paper
  • First Online:
Graph-Theoretic Concepts in Computer Science (WG 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14093))

Included in the following conference series:

Abstract

Graph searches and their respective search trees are widely used in algorithmic graph theory. The problem whether a given spanning tree can be a graph search tree has been considered for different searches, graph classes and search tree paradigms. Similarly, the question whether a particular vertex can be visited last by some search has been studied extensively in recent years. We combine these two problems by considering the question whether a vertex can be a leaf of a graph search tree. We show that for particular search trees, including DFS trees, this problem is easy if we allow the leaf to be the first vertex of the search ordering. We contrast this result by showing that the problem becomes hard for many searches, including DFS and BFS, if we forbid the leaf to be the first vertex. Additionally, we present several structural and algorithmic results for search tree leaves of chordal graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Even on split graphs, the diameter cannot be computed in subquadratic time unless the Strong Exponential Time Hypothesis fails [5].

References

  1. Beisegel, J., et al.: On the end-vertex problem of graph searches. Discret. Math. Theor. Comput. Sci. 21(1) (2019). https://doi.org/10.23638/DMTCS-21-1-13

  2. Beisegel, J., et al.: The recognition problem of graph search trees. SIAM J. Discret. Math. 35(2), 1418–1446 (2021). https://doi.org/10.1137/20M1313301

    Article  MathSciNet  MATH  Google Scholar 

  3. Beisegel, J., Köhler, E., Scheffler, R., Strehler, M.: Linear time LexDFS on chordal graphs. In: Grandoni, F., Herman, G., Sanders, P. (eds.) 28th Annual European Symposium on Algorithms (ESA 2020). LIPIcs, vol. 173, pp. 13:1–13:13. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl (2020). https://doi.org/10.4230/LIPIcs.ESA.2020.13

  4. Berry, A., Blair, J.R., Heggernes, P., Peyton, B.W.: Maximum cardinality search for computing minimal triangulations of graphs. Algorithmica 39(4), 287–298 (2004). https://doi.org/10.1007/s00453-004-1084-3

    Article  MathSciNet  MATH  Google Scholar 

  5. Borassi, M., Crescenzi, P., Habib, M.: Into the square: On the complexity of some quadratic-time solvable problems. In: Crescenzi, P., Loreti, M. (eds.) Proceedings of ICTCS 2015, the 16th Italian Conference on Theoretical Computer Science, ENTCS, vol. 322, pp. 51–67 (2016). https://doi.org/10.1016/j.entcs.2016.03.005

  6. Charbit, P., Habib, M., Mamcarz, A.: Influence of the tie-break rule on the end-vertex problem. Discrete Math. Theor. Comput. Sci. 16(2), 57 (2014). https://doi.org/10.46298/dmtcs.2081

    Article  MathSciNet  MATH  Google Scholar 

  7. Chepoi, V., Dragan, F.: A linear-time algorithm for finding a central vertex of a chordal graph. In: van Leeuwen, J. (ed.) ESA 1994. LNCS, vol. 855, pp. 159–170. Springer, Heidelberg (1994). https://doi.org/10.1007/BFb0049406

    Chapter  Google Scholar 

  8. Corneil, D.G., Dalton, B., Habib, M.: LDFS-based certifying algorithm for the minimum path cover problem on cocomparability graphs. SIAM J. Comput. 42(3), 792–807 (2013). https://doi.org/10.1137/11083856X

    Article  MathSciNet  MATH  Google Scholar 

  9. Corneil, D.G., Dragan, F.F., Habib, M., Paul, C.: Diameter determination on restricted graph families. Discret. Appl. Math. 113(2), 143–166 (2001). https://doi.org/10.1016/S0166-218X(00)00281-X

    Article  MathSciNet  MATH  Google Scholar 

  10. Corneil, D.G., Dusart, J., Habib, M., Mamcarz, A., De Montgolfier, F.: A tie-break model for graph search. Discret. Appl. Math. 199, 89–100 (2016). https://doi.org/10.1016/j.dam.2015.06.011

    Article  MathSciNet  MATH  Google Scholar 

  11. Corneil, D.G., Köhler, E., Lanlignel, J.M.: On end-vertices of lexicographic breadth first searches. Discret. Appl. Math. 158(5), 434–443 (2010). https://doi.org/10.1016/j.dam.2009.10.001

    Article  MathSciNet  MATH  Google Scholar 

  12. Corneil, D.G., Krueger, R.M.: A unified view of graph searching. SIAM J. Discret. Math. 22(4), 1259–1276 (2008). https://doi.org/10.1137/050623498

    Article  MathSciNet  MATH  Google Scholar 

  13. Corneil, D.G., Olariu, S., Stewart, L.: Linear time algorithms for dominating pairs in asteroidal triple-free graphs. SIAM J. Comput. 28(4), 1284–1297 (1999). https://doi.org/10.1137/S0097539795282377

    Article  MathSciNet  MATH  Google Scholar 

  14. Corneil, D.G., Olariu, S., Stewart, L.: The LBFS structure and recognition of interval graphs. SIAM J. Discret. Math. 23(4), 1905–1953 (2009). https://doi.org/10.1137/S0895480100373455

    Article  MathSciNet  MATH  Google Scholar 

  15. Gorzny, J., Huang, J.: End-vertices of LBFS of (AT-free) bigraphs. Discret. Appl. Math. 225, 87–94 (2017). https://doi.org/10.1016/j.dam.2017.02.027

    Article  MathSciNet  MATH  Google Scholar 

  16. Habib, M., McConnell, R., Paul, C., Viennot, L.: Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theoret. Comput. Sci. 234(1–2), 59–84 (2000). https://doi.org/10.1016/S0304-3975(97)00241-7

    Article  MathSciNet  MATH  Google Scholar 

  17. Hagerup, T.: Biconnected graph assembly and recognition of DFS trees. Technical Report, A 85/03, Universität des Saarlandes (1985). https://doi.org/10.22028/D291-26437

  18. Hagerup, T., Nowak, M.: Recognition of spanning trees defined by graph searches. Technical Report, A 85/08, Universität des Saarlandes (1985)

    Google Scholar 

  19. Hopcroft, J., Tarjan, R.E.: Efficient planarity testing. J. ACM 21(4), 549–568 (1974). https://doi.org/10.1145/321850.321852

    Article  MathSciNet  MATH  Google Scholar 

  20. Korach, E., Ostfeld, Z.: DFS tree construction: algorithms and characterizations. In: van Leeuwen, J. (ed.) WG 1988. LNCS, vol. 344, pp. 87–106. Springer, Heidelberg (1989). https://doi.org/10.1007/3-540-50728-0_37

    Chapter  Google Scholar 

  21. Kratsch, D., Damaschke, P., Lubiw, A.: Dominating cliques in chordal graphs. Discret. Math. 128(1), 269–275 (1994). https://doi.org/10.1016/0012-365X(94)90118-X

    Article  MathSciNet  MATH  Google Scholar 

  22. Kratsch, D., Liedloff, M., Meister, D.: End-vertices of graph search algorithms. In: Paschos, V.T., Widmayer, P. (eds.) CIAC 2015. LNCS, vol. 9079, pp. 300–312. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-18173-8_22

    Chapter  MATH  Google Scholar 

  23. Krueger, R., Simonet, G., Berry, A.: A general label search to investigate classical graph search algorithms. Discret. Appl. Math. 159(2–3), 128–142 (2011). https://doi.org/10.1016/j.dam.2010.02.011

    Article  MathSciNet  MATH  Google Scholar 

  24. Manber, U.: Recognizing breadth-first search trees in linear time. Inf. Process. Lett. 34(4), 167–171 (1990). https://doi.org/10.1016/0020-0190(90)90155-Q

    Article  MathSciNet  MATH  Google Scholar 

  25. Rong, G., Cao, Y., Wang, J., Wang, Z.: Graph searches and their end vertices. Algorithmica 84, 2642–2666 (2022). https://doi.org/10.1007/s00453-022-00981-5

    Article  MathSciNet  MATH  Google Scholar 

  26. Rose, D.J.: Triangulated graphs and the elimination process. J. Math. Anal. Appl. 32(3), 597–609 (1970). https://doi.org/10.1016/0022-247X(70)90282-9

    Article  MathSciNet  MATH  Google Scholar 

  27. Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976). https://doi.org/10.1137/0205021

    Article  MathSciNet  MATH  Google Scholar 

  28. Scheffler, R.: Linearizing partial search orders. In: Bekos, M.A., Kaufmann, M. (eds.) WG 2022. LNCS, vol. 13453, pp. 425–438. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-15914-5_31

    Chapter  Google Scholar 

  29. Scheffler, R.: On the recognition of search trees generated by BFS and DFS. Theoret. Comput. Sci. 936, 116–128 (2022). https://doi.org/10.1016/j.tcs.2022.09.018

    Article  MathSciNet  MATH  Google Scholar 

  30. Scheffler, R.: Graph search trees and their leaves. Preprint on arXiv (2023). https://doi.org/10.48550/arXiv.2307.07279

  31. Spinrad, J., Sritharan, R.: Algorithms for weakly triangulated graphs. Discret. Appl. Math. 59(2), 181–191 (1995). https://doi.org/10.1016/0166-218X(93)E0161-Q

    Article  MathSciNet  MATH  Google Scholar 

  32. Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13(3), 566–579 (1984). https://doi.org/10.1137/0213035

    Article  MathSciNet  MATH  Google Scholar 

  33. Zou, M., Wang, Z., Wang, J., Cao, Y.: End vertices of graph searches on bipartite graphs. Inf. Proc. Lett. 173, 106176 (2022). https://doi.org/10.1016/j.ipl.2021.106176

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Brandenburg University of Technology, Cottbus, Germany

    Robert Scheffler

Authors
  1. Robert Scheffler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Scheffler .

Editor information

Editors and Affiliations

  1. Durham University, Durham, UK

    Daniël Paulusma

  2. University of Fribourg, Fribourg, Fribourg, Switzerland

    Bernard Ries

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Scheffler, R. (2023). Graph Search Trees and Their Leaves. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_33

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-43380-1_33

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-43379-5

  • Online ISBN: 978-3-031-43380-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation