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Graph Searches and Their End Vertices

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Abstract

For a graph search algorithm, the end vertex problem is concerned with which vertices of a graph can be the last visited by this algorithm. We characterize all maximum cardinality searches on chordal graphs and derive from this characterization a polynomial-time algorithm for the end vertex problem of maximum cardinality searches on chordal graphs. 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.

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Notes

  1. A graph is a split graph if its vertex set can be partitioned into a clique and an independent set. Definitions of weakly chordal graphs and interval graphs are deferred to Sects. 4 and 6, respectively.

  2. The algorithm mcs\(^{+}\) is defined in the same spirit of lbfs\(^{+}\), and it has not been explicitly mentioned in literature because no application of this algorithm has been discovered.

  3. This is not a problem for lbfs\(^{+}\), because it never merges two parts.

  4. One may note that \(\{[{\mathtt {lp}(v)}, {\mathtt {rp}(v)}]\mid v\in V(G)\}\) gives an interval representation for G.

References

  1. Beisegel, J., Denkert, C., Köhler, E., Krnc, M., Pivac, N., Scheffler, R., Strehler, M.: On the end-vertex problem of graph searches. Discrete Math. Theor. Comput. Sci. 21(1), (2019). https://doi.org/10.23638/DMTCS-21-1-13

  2. Bernstein, P.A., Goodman, N.: Power of natural semijoins. SIAM J. Comput. 10(4), 751–771 (1981). https://doi.org/10.1137/0210059

    Article  MathSciNet  MATH  Google Scholar 

  3. Berry, A., Blair, J.R.S., Bordat, J.P., Simonet, G.: Graph extremities defined by search algorithms. Algorithms 3(2), 100–124 (2010). https://doi.org/10.3390/a3020100

    Article  MathSciNet  MATH  Google Scholar 

  4. Berry, A.: Separability generalizes Dirac’s theorem. Discrete Appl. Math. 84(1–3), 43–53 (1998). https://doi.org/10.1016/S0166-218X(98)00005-5

    Article  MathSciNet  MATH  Google Scholar 

  5. Blair, J.R.S., Peyton, B.W.: An introduction to chordal graphs and clique trees. In: George, J.A., Gilbert, J.R., Liu, J.W.-H. (eds.) Graph Theory and Sparse Matrix Computation, Volume 56 of IMA, pp. 1–29. Springer, Berlin (1993)

    Google Scholar 

  6. Cao, Y.: Recognizing (unit) interval graphs by zigzag graph searches. In: Le Viet, H., King, V. (eds) Proceedings of the 4th SIAM Symposium on Simplicity in Algorithms (SOSA), pp. 92–106. SIAM (2021). https://doi.org/10.1137/1.9781611976496.11

  7. Charbit, P., Habib, M., Mamcarz, A.: Influence of the tie-break rule on the end-vertex problem. Discrete Math. Theor. Comput. Sci. 16(2), 57–72 (2014). https://doi.org/10.46298/dmtcs.2081

    Article  MathSciNet  MATH  Google Scholar 

  8. Corneil, D.G.: Lexicographic breadth first search: a survey. In: Volume 3353 of LNCS, pp. 1–19. Springer (2004). https://doi.org/10.1007/978-3-540-30559-0_1

  9. Corneil, D.G.: A simple 3-sweep LBFS algorithm for the recognition of unit interval graphs. Discrete Appl. Math. 138(3), 371–379 (2004). https://doi.org/10.1016/j.dam.2003.07.001

    Article  MathSciNet  MATH  Google Scholar 

  10. 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 

  11. Corneil, D.G., Köhler, E., Lanlignel, J.-M.: On end-vertices of lexicographic breadth first searches. Discrete 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.: A unified view of graph searching. SIAM J. Discrete 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. (A preliminary version appeared in ICALP 1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Corneil, D.G., Olariu, S., Stewart, L.: The LBFS structure and recognition of interval graphs. SIAM J. Discrete Math. 23(4), 1905–1953 (2009). https://doi.org/10.1137/S0895480100373455. (A preliminary version appeared in SODA 1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dirac, G.A.: On rigid circuit graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 25(1), 71–76 (1961). https://doi.org/10.1007/BF02992776

    Article  MathSciNet  MATH  Google Scholar 

  16. Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pac. J. Math. 15(3), 835–855 (1965). https://doi.org/10.2140/pjm.1965.15.835

    Article  MathSciNet  MATH  Google Scholar 

  17. Galinier, P., Habib, M., Paul, C.: Chordal graphs and their clique graphs. In: Nagl, M. (eds) Proceedings of the 21st International Workshop on Graph-Theoretic Concepts in Computer Science (WG), volume 1017 of LNCS, pp. 358–371. Springer (1995). https://doi.org/10.1007/3-540-60618-1_88

  18. Habib, M., McConnell, R.M., 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 

  19. Hopcroft, J.E., 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. Impagliazzo, R., Paturi, R.: On the complexity of \(k\)-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001). https://doi.org/10.1006/jcss.2000.1727. (A preliminary version appeared in CCC 1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kratsch, D., Liedloff, M., Meister, D.: End-vertices of graph search algorithms. In: Paschos, V.T., Widmayer, P. (eds) Proceedings of the 12th International Conference on Algorithms and Complexity (CIAC), volume 9079 of Lecture Notes in Computer Science, pp. 300–312. Springer (2015). https://doi.org/10.1007/978-3-319-18173-8_22

  22. Li, P., Yaokun, W.: A four-sweep LBFS recognition algorithm for interval graphs. Discrete Math. Theor. Comput. Sci. 16(3), 23–50 (2014). https://doi.org/10.46298/dmtcs.2094

    Article  MathSciNet  MATH  Google Scholar 

  23. Nagamochi, H.: Computing edge-connectivity in multigraphs and capacitated graphs. SIAM J. Discrete Math. 5(1), 54–66 (1992). https://doi.org/10.1137/0405004

    Article  MathSciNet  MATH  Google Scholar 

  24. Nagamochi, H., Ibaraki, T.: Algorithmic aspects of graph connectivity. In: Encyclopedia of Mathematics and its Applications. Cambridge University Press (2008). https://doi.org/10.1017/CBO9780511721649

  25. 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. (A preliminary version appeared in STOC 1975)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sethi, R.: Scheduling graphs on two processors. SIAM J. Comput. 5(1), 73–82 (1976). https://doi.org/10.1137/0205005

    Article  MathSciNet  MATH  Google Scholar 

  27. Shier, D.R.: Some aspects of perfect elimination orderings in chordal graphs. Discrete Appl. Math. 7(3), 325–331 (1984). https://doi.org/10.1016/0166-218X(84)90008-8

    Article  MathSciNet  MATH  Google Scholar 

  28. Simon, K.: 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, pp. 289–308 (1991). https://doi.org/10.1007/3-540-54891-2_22

  29. Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972). https://doi.org/10.1137/0201010. (A preliminary version appeared in SWAT (FOCS) 1971)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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). With Addendum in the same Journal 14(1):254–255 (1985.) https://doi.org/10.1137/0213035

  31. Tedder, M., Corneil, D.G., Habib, M., Paul, C.: Simpler linear-time modular decomposition via recursive factorizing permutations. In: Automata, Languages and Programming (ICALP), Volume 5125 of LNCS, pp. 634–645. Berlin, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70575-8_52

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Y.C. would like to thank Jing Huang for bringing the end vertex problems to his attention.

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Authors and Affiliations

  1. School of Computer Science and Engineering, Central South University, Changsha, China

    Guozhen Rong, Yixin Cao, Jianxin Wang & Zhifeng Wang

  2. Department of Computing, Hong Kong Polytechnic University, Hong Kong, China

    Yixin Cao

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Correspondence to Yixin Cao.

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A preliminary version of this paper appears in the Proceedings of the 30th International Symposium on Algorithms and Computation (ISAAC 2019).

G. Rong, Y. Cao, J. Wang, Z. Wang: Supported by National Natural Science Foundation of China (61828205, 61672536), Hunan Provincial Key Lab on Bioinformatics, and Hunan Provincial Science and Technology Program (2018WK4001).

Y. Cao: Supported in part by the Hong Kong Research Grants Council (RGC) under Grant 15201317 and the National Natural Science Foundation of China (NSFC) under Grant 61972330.

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Rong, G., Cao, Y., Wang, J. et al. Graph Searches and Their End Vertices. Algorithmica 84, 2642–2666 (2022). https://doi.org/10.1007/s00453-022-00981-5

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