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End-Vertices of AT-free Bigraphs

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Computing and Combinatorics (COCOON 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12273))

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Abstract

The end-vertex problem for a search algorithm asks whether a vertex of the input graph is the last visited vertex of an execution of that search algorithm. We consider the end-vertex problem restricted to AT-free bigraphs for various search algorithms: Breadth-First Search (BFS), Lexicographic Breadth-First Search (LBFS), Depth-First Search (DFS), and Maximal Neighbourhood Search (MNS). Deciding whether a vertex of a graph is the end-vertex of any of these search algorithms is NP-complete in general. We show that we can decide whether a vertex is an end-vertex of BFS or MNS in polynomial time on AT-free bigraphs. Additionally, we show that we can decide whether a vertex is an end-vertex of DFS or LBFS in linear time on AT-free bigraphs; this improves the LBFS end-vertex complexity on this class of graphs.

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References

  1. Beisegel, J.: On the end-vertex problem of graph searches. Discret. Math. Theor. Comput. Sci. 21(1) (2019). http://dmtcs.episciences.org/5572

  2. Berry, A., Blair, J.R.S., Bordat, J.-P., Simonet, G.: Graph extremities defined by search algorithms. Algorithms 3(2), 100–124 (2010)

    Article  MathSciNet  Google Scholar 

  3. Berry, A., Bordat, J.-P.: Local LexBFS properties in an arbitrary graph. In: Proceedings of Journées Informatiques Messines (JIM) (2000)

    Google Scholar 

  4. Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, Cham (2008)

    Book  Google Scholar 

  5. Brandstädt, A., Spinard, J.P., Bang Le, V.: Graph Classes: A Survey, vol. 3. SIAM (1999)

    Google Scholar 

  6. Cao, Y., Wang, Z., Rong, G., Wang, J.: Graph searches and their end vertices. In: Lu, P., Zhang, G. (eds.) 30th International Symposium on Algorithms and Computation, (ISAAC), LIPIcs, vol. 149, pp. 1:1–1:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019). https://doi.org/10.4230/LIPIcs.ISAAC.2019.1

  7. Chang, J.-M., Ho, C.-W., Ko, M.-T.: LexBFS-ordering in asteroidal triple-free graphs. ISAAC 1999. LNCS, vol. 1741, pp. 163–172. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-46632-0_17

    Chapter  Google Scholar 

  8. Charbit, P., Habib, M., Mamcarz, A.: Influence of the tie-break rule on the end-vertex problem (2014)

    Google Scholar 

  9. Corneil, D.G.: Lexicographic breadth first search – a survey. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 1–19. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30559-0_1

    Chapter  Google Scholar 

  10. Corneil, D.G., Dragan, F.F., Habib, M., Paul, C.: Diameter determination on restricted graph families. Discret. Appl. Math. 113(2–3), 143–166 (2001)

    Article  MathSciNet  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  Google Scholar 

  15. Dragan, F.F., Nicolai, F.: LexBFS orderings of distance hereditary graphs. UD, Fachbereich Mathematik (1995)

    Google Scholar 

  16. Gorzny, J.: On end vertices of search algorithms. Master’s thesis, University of Victoria, Victoria, BC, Canada (2015)

    Google Scholar 

  17. Gorzny, J., Huang, J.: End-vertices of LBFS of (AT-free) bigraphs. Discret. Appl. Math. 225, 87–94 (2017)

    Article  MathSciNet  Google Scholar 

  18. Hell, P., Huang, J.: Interval bigraphs and circular arc graphs. J. Graph Theory 46(4), 313–327 (2004)

    Article  MathSciNet  Google Scholar 

  19. Jamison, B., Olariu, S.: On the semi-perfect elimination. Adv. Appl. Math. 9(3), 364–376 (1988)

    Article  MathSciNet  Google Scholar 

  20. 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 

  21. Lekkerkerker, C., Boland, J.: Representation of a finite graph by a set of intervals on the real line. Fundamenta Mathematicae 51(1), 45–64 (1962)

    Article  MathSciNet  Google Scholar 

  22. Mertzios, G.B., Nichterlein, A., Niedermeier, R.: A linear-time algorithm for maximum-cardinality matching on cocomparability graphs. SIAM J. Discret. Math. 32(4), 2820–2835 (2018). https://doi.org/10.1137/17M1120920

    Article  MathSciNet  MATH  Google Scholar 

  23. Donald Rose, J., Endre Tarjan, R., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)

    Article  MathSciNet  Google Scholar 

  24. Spinrad, J., Brandstädt, A., Stewart, L.: Bipartite permutation graphs. Discret. Appl. Math. 18(3), 279–292 (1987)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University of Waterloo, Waterloo, ON, Canada

    Jan Gorzny

  2. University of Victoria, Victoria, BC, Canada

    Jing Huang

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  1. Jan Gorzny
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  2. Jing Huang
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Correspondence to Jan Gorzny .

Editor information

Editors and Affiliations

  1. Georgia State University, Atlanta, USA

    Donghyun Kim

  2. Department of Mathematics and Physics, North Carolina Central University, Durham, NC, USA

    R. N. Uma

  3. Computer Science, Georgia State University, Atlanta, GA, USA

    Zhipeng Cai

  4. Korea University, Seoul, Korea (Republic of)

    Dong Hoon Lee

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Gorzny, J., Huang, J. (2020). End-Vertices of AT-free Bigraphs. In: Kim, D., Uma, R., Cai, Z., Lee, D. (eds) Computing and Combinatorics. COCOON 2020. Lecture Notes in Computer Science(), vol 12273. Springer, Cham. https://doi.org/10.1007/978-3-030-58150-3_5

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  • DOI: https://doi.org/10.1007/978-3-030-58150-3_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-58149-7

  • Online ISBN: 978-3-030-58150-3

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