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A Polynomial-Time Algorithm for MCS Partial Search Order on Chordal Graphs

Authors Guozhen Rong, Yongjie Yang , Wenjun Li

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Author Details

Guozhen Rong
  • Hunan Provincial Key Laboratory of Intelligent Processing of Big Data on Transportation, Changsha University of Science and Technology, China
Yongjie Yang
  • Chair of Economic Theory, Saarland University, Saarbrücken, Germany
Wenjun Li
  • Hunan Provincial Key Laboratory of Intelligent Processing of Big Data on Transportation, Changsha University of Science and Technology, China

Funding

This paper was supported by the Research Foundation of Education Bureau of Hunan Province under grant 21B0305 and Natural Science Foundation of Hunan Province of China under grant 2022JJ30620.

Acknowledgements

The authors thank the anonymous reviewers of MFCS 2023 for their careful reading and instructive comments.

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Guozhen Rong, Yongjie Yang, and Wenjun Li. A Polynomial-Time Algorithm for MCS Partial Search Order on Chordal Graphs. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 77:1-77:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.MFCS.2023.77

Abstract

We study the partial search order problem (PSOP) proposed recently by Scheffler [WG 2022]. Given a graph G together with a partial order on the set of vertices of G, this problem determines if there is an 𝒮-ordering that is consistent with the given partial order, where 𝒮 is a graph search paradigm like BFS, DFS, etc. This problem naturally generalizes the end-vertex problem which has received much attention over the past few years. It also generalizes the so-called ℱ-tree recognition problem which has just been studied in the literature recently. Our main contribution is a polynomial-time dynamic programming algorithm for the PSOP of the maximum cardinality search (MCS) restricted to chordal graphs. This resolves one of the most intriguing open questions left in the work of Scheffler [WG 2022]. To obtain our result, we propose the notion of layer structure and study numerous related structural properties which might be of independent interest.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • partial search order
  • maximum cardinality search
  • chordal graphs
  • clique graphs
  • dynamic programming

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References

  1. Jesse Beisegel, Carolin Denkert, Ekkehard Köhler, Matjaž Krnc, Nevena Pivač, Robert Scheffler, and Martin Strehler. On the end-vertex problem of graph searches. Discrete Mathematics & Theoretical Computer Science, 21(1):Nr. 13, 2019. URL: https://doi.org/10.23638/DMTCS-21-1-13.
  2. Jesse Beisegel, Carolin Denkert, Ekkehard Köhler, Matjaž Krnc, Nevena Pivač, Robert Scheffler, and Martin Strehler. The recognition problem of graph search trees. SIAM Journal on Discrete Mathematics, 35(2):1418-1446, 2021. URL: https://doi.org/10.1137/20M1313301.
  3. Anne Berry, Jean R. S. Blair, Pinar Heggernes, and Barry W. Peyton. Maximum cardinality search for computing minimal triangulations of graphs. Algorithmica, 39(4):287-298, 2004. URL: https://doi.org/10.1007/s00453-004-1084-3.
  4. Jean R. S. Blair and Barry W. Peyton. An introduction to chordal graphs and clique trees. In Graph Theory and Sparse Matrix Computation, pages 1-29. Springer, 1993. URL: https://doi.org/10.1007/978-1-4613-8369-7_1.
  5. Hans L. Bodlaender and Arie M. C. A. Koster. On the maximum cardinality search lower bound for treewidth. Discrete Applied Mathematics, 155(11):1348-1372, 2007. URL: https://doi.org/10.1016/j.dam.2007.02.004.
  6. Anna Bretscher, Derek Gordon Corneil, Michel Habib, and Christophe Paul. A simple linear time LexBFS cograph recognition algorithm. SIAM Journal on Discrete Mathematics, 22(4):1277-1296, 2008. URL: https://doi.org/10.1137/060664690.
  7. Peter Buneman. A characterisation of rigid circuit graphs. Discrete Mathematics, 9(3):205-212, 1974. URL: https://doi.org/10.1016/0012-365X(74)90002-8.
  8. Derek Gordon 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: https://doi.org/10.1137/11083856X.
  9. Derek Gordon Corneil, Jérémie Dusart, Michel Habib, and Ekkehard Köhler. On the power of graph searching for cocomparability graphs. SIAM Journal on Discrete Mathematics, 30(1):569-591, 2016. URL: https://doi.org/10.1137/15M1012396.
  10. Derek Gordon Corneil and Richard Krueger. A unified view of graph searching. SIAM Journal on Discrete Mathematics, 22(4):1259-1276, 2008. URL: https://doi.org/10.1137/050623498.
  11. Derek Gordon Corneil, Ekkehard Köhler, and Jean-Marc Lanlignel. On end-vertices of lexicographic breadth first searches. Discrete Applied Mathematics, 158(5):434-443, 2010. URL: https://doi.org/10.1016/j.dam.2009.10.001.
  12. Edsger Wybe Dijkstra. A note on two problems in connexion with graphs. Numerische Mathematik, 1(1):269-271, 1959. URL: https://doi.org/10.1007/BF01386390.
  13. Gabriel Andrew Dirac. On rigid circuit graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 25:71-76, 1961. URL: https://doi.org/10.1007/BF02992776.
  14. Philippe Galinier, Michel Habib, and Christophe Paul. Chordal graphs and their clique graphs. In Manfred Nagl, editor, WG, volume 1017 of Lecture Notes in Computer Science, pages 358-371. Springer, 1995. URL: https://doi.org/10.1007/3-540-60618-1_88.
  15. F̌aniča Gavril. The intersection graphs of subtrees in trees are exactly the chordal graphs. Journal of Combinatorial Theory, Series B, 16(1):47-56, 1974. URL: https://doi.org/10.1016/0095-8956(74)90094-X.
  16. Jan Gorzny. Related Orderings of AT-Free Graphs. PhD thesis, University of Waterloo, 2022. URL: https://uwspace.uwaterloo.ca/handle/10012/18079.
  17. Torben Hagerup. Biconnected graph assembly and recognition of DFS trees. Technical report, Universität des Saarlandes, 1985. URL: https://doi.org/10.22028/D291-26437.
  18. Torben Hagerup and Manfred Nowak. Recognition of spanning trees defined by graph searches. Technical report, Universität des Saarlandes, 1985. Google Scholar
  19. Vojtěch Jarník. O jistém problému minimálním. Práce Moravské Přírodovědecké Společnosti, 6(4):57-63, 1930. URL: https://dml.cz/bitstream/handle/10338.dmlcz/500726/Jarnik_01-0000-31_1.pdf.
  20. Ephraim Korach and Zvi Ostfeld. DFS tree construction: Algorithms and characterizations. In Jan van Leeuwen, editor, WG, volume 344 of Lecture Notes in Computer Science, pages 87-106. Springer, 1988. URL: https://doi.org/10.1007/3-540-50728-0_37.
  21. Arie M. C. A. Koster, Hans L. Bodlaender, and Stan P. M. van Hoesel. Treewidth: Computational experiments. Electronic Notes in Discrete Mathematics, 8:54-57, 2001. URL: https://doi.org/10.1016/S1571-0653(05)80078-2.
  22. P. Sreenivasa Kumar and C. E. Veni Madhavan. Minimal vertex separators of chordal graphs. Discrete Applied Mathematics, 89(1-3):155-168, 1998. URL: https://doi.org/10.1016/S0166-218X(98)00123-1.
  23. Robert Clay Prim. Shortest connection networks and some generalizations. Bell System Technical Journal, 36(6):1389-1401, 1957. URL: https://doi.org/10.1002/j.1538-7305.1957.tb01515.x.
  24. Guozhen Rong, Yixin Cao, Jianxin Wang, and Zhifeng Wang. Graph searches and their end vertices. Algorithmica, 84(9):2642-2666, 2022. URL: https://doi.org/10.1007/s00453-022-00981-5.
  25. 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. URL: https://doi.org/10.1137/0205021.
  26. Robert Scheffler. Linearizing partial search orders. In Michael A. Bekos and Michael Kaufmann, editors, WG, volume 13453 of Lecture Notes in Computer Science, pages 425-438. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-15914-5_31.
  27. Robert Scheffler. On the recognition of search trees generated by BFS and DFS. Theoretical Computer Science, 936:116-128, 2022. URL: https://doi.org/10.1016/j.tcs.2022.09.018.
  28. Robert Endre Tarjan. Depth-first search and linear graph algorithms. SIAM Journal on Computing, 1(2):146-160, 1972. URL: https://doi.org/10.1137/0201010.
  29. 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. URL: https://doi.org/10.1137/0213035.
  30. James Richard Walte. Representations of Rigid Cycle Graphs. PhD thesis, Wayne State University, 1972. Google Scholar
  31. Douglas B. West. Introduction to Graph Theory. Prentice Hall, second edition, 2001. URL: https://openlibrary.org/books/OL6785828M/Introduction_to_graph_theory.
  32. Meibiao Zou, Zhifeng Wang, Jianxin Wang, and Yixin Cao. End vertices of graph searches on bipartite graphs. Information Processing Letters, 173:Nr. 106176, 2022. URL: https://doi.org/10.1016/j.ipl.2021.106176.
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