Skip to main content
SpringerLink
Log in

Acyclic Digraphs

  • Chapter
  • First Online:
Book cover Classes of Directed Graphs

Part of the book series: Springer Monographs in Mathematics ((SMM))

Abstract

A digraph is acyclic if it has no dicycle. Acyclic digraphs form a well-studied family of digraphs of great interest in graph theory, algorithms and applications. We consider some basic results on acyclic digraphs and introduce transitive digraphs, and the transitive closure and transitive reduction of a digraph. We discuss results on out- and in-branchings, the k-linkage problem, maximum dicuts, and the multicut problem. We present enumeration results for acyclic digraphs, and results on maximum spanning and induced subgraphs of digraphs. Four sections are devoted to applications of acyclic digraphs: in embedded computing, cryptographic enforcement schemes, project schedulting, and text analysing. The final section is on acyclic edge-coloured graphs which generalize acyclic digraphs.

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 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 89.99
Price excludes VAT (USA)
  • Durable hardcover 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.

    In this chapter, we consider only some applications of acyclic digraphs. There are many others and some even appeared while the chapter was being written, see e.g. [6], where Antoniou, Araújo, Bustamante and Gibali used basic properties of acyclic digraphs to design an algorithm for disassembling toy models produced by Engino\(^{\textregistered }\).

  2. 2.

    Note that in this parameterization k is not necessarily an integer, but an integer divided by 2, such that \(W/2+k\) is an integer. A similar remark also holds for the other parameterization of this section.

  3. 3.

    There is an arc version of Multicut, where arcs are to separate the terminal pairs of vertices. However, the vertex and arc versions have the same classical and parameterized complexity for digraphs [20].

  4. 4.

    We can clearly reduce \(O(|A'| + n^2)\) to \(O(n^2)\); the reason we keep \(|A'|\) is to stress that we can consider only arcs of H.

  5. 5.

    For the basics on network flows, see Section 1.9 of Chapter 1.

  6. 6.

    We can assign bounds to vertices rather than arcs as every vertex can be split, i.e., replaced by an arc, see Section 1.4 of Chapter 1.

  7. 7.

    The original versions of PERT and CPM used another type of network, activity-on-arc (AOA) project networks, but AOA networks are significantly harder to construct and change than AON networks and it makes more sense to use AON networks rather than AOA networks.

  8. 8.

    The idea of using FPT algorithms to evaluate heuristics was likely coined first by Gutin, Karapetyan and Razgon [43].

References

  1. A. Abouelaoualim, K.Ch. Das, L. Faria, Y. Manoussakis, C.A. Martinhon, and R. Saad. Paths and trails in edge-colored graphs. Theor. Comput. Sci., 409:497–510, 2008.

    Article  MathSciNet  Google Scholar 

  2. A.V. Aho, M.R. Garey, and J.D. Ullman. The transitive reduction of a directed graph. SIAM J. Comput., 1(2):131–137, 1972.

    Article  MathSciNet  Google Scholar 

  3. S. Alamdari and A. Mehrabian. On a DAG partitioning problem. In WAW 2012, volume 7323 of Lect. Notes Comput. Sci., pages 17–28. Springer, 2012.

    Google Scholar 

  4. N. Alon, B. Bollobàs, A. Gyàrfàs, J. Lehel, and A. Scott. Maximum directed cuts in acyclic digraphs. J. Graph Theory, 55:1–13, 2007.

    Article  MathSciNet  Google Scholar 

  5. N. Alon, F.V. Fomin, G. Gutin, M. Krivelevich, and S. Saurabh. Spanning directed trees with many leaves. SIAM J. Discrete Math., 23(1):466–476, 2009.

    Article  MathSciNet  Google Scholar 

  6. E.N. Antoniou, A. Araújo, M.D. Bustamante, and A. Gibali. Physically feasible decomposition of Engino\(^{\textregistered }\) toy models: A graph theoretic approach. CoRR, abs/1707.09040, 2017.

    Google Scholar 

  7. P. Balister, S. Gerke, G. Gutin, A. Johnstone, J. Reddington, E. Scott, A. Soleimanfallah, and A. Yeo. Algorithms for generating convex sets in acyclic digraphs. J. Discrete Algor., 7:509–518, 2009.

    Article  MathSciNet  Google Scholar 

  8. J. Bang-Jensen and S. Bessy. (Arc-)disjoint flows in networks. Theor. Comput. Sci., 526:28–40, 2014.

    Article  MathSciNet  Google Scholar 

  9. J. Bang-Jensen and G. Gutin. Digraphs: Theory, Algorithms and Applications. Springer-Verlag, London, 2nd edition, 2009.

    Book  Google Scholar 

  10. J. Bang-Jensen and G. Gutin. Out-branchings with extremal number of leaves. Ramanujan Math. Soc. Lect. Notes, 13:91–99, 2010.

    MathSciNet  MATH  Google Scholar 

  11. J. Bang-Jensen and M. Kriesell. Disjoint directed and undirected paths and cycles in digraphs. Theor. Comput. Sci., 410:5138–5144, 2009.

    Article  MathSciNet  Google Scholar 

  12. J. Bang-Jensen, S. Thomassé, and A. Yeo. Small degree out-branchings. J. Graph Theory, 42(4):297–307, 2003.

    Article  MathSciNet  Google Scholar 

  13. J. Bang-Jensen and A. Yeo. The complexity of multicut and mixed multicut problems in (di)graphs. Theor. Comput. Sci., 520, 2014.

    Article  MathSciNet  Google Scholar 

  14. K. Bérczi, S. Fujishige, and N. Kamiyama. A linear-time algorithm to find a pair of arc-disjoint spanning in-arborescence and out-arborescence in a directed acyclic graph. Inform. Process. Lett., 109(23-24):1227–1231, 2009.

    Article  MathSciNet  Google Scholar 

  15. D. Bokal, G. Fijavz, M. Juvan, P.M. Kayll, and B. Mohar. The circular chromatic number of a digraph. J. Graph Theory, 46(3):227–240, 2004.

    Article  MathSciNet  Google Scholar 

  16. P. Bonzini and L. Pozzi. Polynomial-time subgraph enumeration for automated instruction set extension. In DATE 2007: Conference on Design, Automation and Test in Europe, pages 1331–1336, 2007.

    Google Scholar 

  17. J. Chen, Y. Liu, S. Lu, B. O’Sullivan, and I. Razgon. A fixed-parameter algorithm for the directed feedback vertex set problem. J. Assoc. Comput. Mach., 55(5):21:1–21:19, 2008.

    Article  MathSciNet  Google Scholar 

  18. J. Chen, Y. Liu, S. Lu, B. O’Sullivan, and I. Razgon. Directed Feedback Vertex Set is Fixed-Parameter Tractable. In STOC 2008: 40th Annual ACM Symposium on Theory of Computing. ACM Press, 2008.

    Google Scholar 

  19. X. Chen, D.L. Maskell, and Y. Sun. Fast identification of custom instructions for extensible processors. IEEE Trans. Computer-Aided Design Integr. Circuits Syst., 26:359–368, 2007.

    Article  Google Scholar 

  20. R.H. Chitnis, M. Hajiaghayi, and D. Marx. Fixed-parameter tractability of directed multiway cut parameterized by the size of the cutset. SIAM J. Comput., 42(4):1674–1696, 2013.

    Article  MathSciNet  Google Scholar 

  21. D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progressions. In STOC 1987: 19th Ann. ACM Symp. on Theory of Computation, pages 1–6, ACM Press, 1987.

    Google Scholar 

  22. J. Crampton, R. Daud, and K.M. Martin. Constructing key assignment schemes from chain partitions. In DBSEC 2010: the 24th Conference on Data and Applications Security and Privacy, pages 130–145, 2010.

    Google Scholar 

  23. J. Crampton, N. Farley, G. Gutin, and M. Jones. Optimal constructions for chain-based cryptographic enforcement of information flow policies. In DBSec 2015, volume 9149 of Lect. Notes Comput. Sci., pages 330–345. Springer, 2015.

    Google Scholar 

  24. J. Crampton, N. Farley, G. Gutin, M. Jones, and B. Poettering. Cryptographic enforcement of information flow policies without public information. In ACNS 2015, volume 9092 of Lect. Notes Comput. Sci., pages 389–408. Springer, 2015.

    Google Scholar 

  25. J. Crampton, N. Farley, G. Gutin, M. Jones, and B. Poettering. Cryptographic enforcement of information flow policies without public information via tree partitions. J. Comput. Security, 25:511–535, 2017.

    Article  Google Scholar 

  26. R. Crowston, G. Gutin, and M. Jones. Directed acyclic subgraph problem parameterized above the Poljak-Turzik bound. In FSTTCS 2012, volume 18 of LIPIcs, pages 400–411, 2012.

    Google Scholar 

  27. J. Daligault and S. Thomassé. On finding directed trees with many leaves. In IWPEC 2009, volume 5917 of Lect. Notes Comput. Sci., pages 86–97. Springer, 2009.

    Google Scholar 

  28. A. Demers and A. Downing. Minimum leaf spanning tree. US Patent no. 6,105,018, August 2000.

    Google Scholar 

  29. R.P. Dilworth. A decomposition theorem for partially ordered sets. Ann. Math., 51:161–166, 1950.

    Article  MathSciNet  Google Scholar 

  30. J. Edmonds. Optimum branchings. J. Res. Natl. Bur. Stand. Sect. B, 71B:233–240, 1967.

    Article  MathSciNet  Google Scholar 

  31. C.S. Edwards. Some extremal properties of bipartite subgraphs. Can. J. Math., 25:475–485, 1973.

    Article  MathSciNet  Google Scholar 

  32. M.J. Fisher and A.R. Meyer. Boolean matrix multiplication and transitive closure. In 12th Ann. ACM Symp. on Switching and Automata Theory, pages 129–131. ACM Press, 1971.

    Google Scholar 

  33. S. Fortune, J.E. Hopcroft, and J. Wyllie. The directed subgraph homeomorphism problem. Theor. Comput. Sci., 10:111–121, 1980.

    Article  MathSciNet  Google Scholar 

  34. S. Fujita and C. Magnant. Properly colored paths and cycles. Discrete Appl. Math., 159:1391–1397, 2011.

    Article  MathSciNet  Google Scholar 

  35. M.E. Furman. Application of a method of fast multiplication of matrices in the problem of finding the transitive closure of a graph. Sov. Math. Dokl., 11:1252, 1970.

    MATH  Google Scholar 

  36. I.M. Gessel. Counting acyclic digraphs by sources and sinks. Discrete Math., 160:253–258, 1996.

    Article  MathSciNet  Google Scholar 

  37. N. Golowich and D. Rolnick. Acyclic subgraphs of planar digraphs. Electr. J. Combin., 22(3):P3.7, 2015.

    Google Scholar 

  38. A. Goralcikova and V. Koubek. A reduct-and-closure algorithm for graphs. In MFCS 1979: 8th Symp. on Math. Foundations of Computer Science, volume 74 of Lect. Notes Comput. Sci., pages 301–307. Springer-Verlag, 1979.

    Google Scholar 

  39. G. Gutin. A note on the cardinality of certain classes of unlabeled multipartite tournaments. Discrete Math., 186(1-3):277–280, 1998.

    Article  MathSciNet  Google Scholar 

  40. G. Gutin, A. Johnstone, J. Reddington, E. Scott, and A. Yeo. An algorithm for finding input/output constrained convex sets in an acyclic digraph. J. Discrete Algor., 13:47–58, 2012.

    Article  MathSciNet  Google Scholar 

  41. G. Gutin, M. Jones, B. Sheng, M. Wahlström, and A. Yeo. Acyclicity in edge-colored graphs. Discrete Math., 340:1–8, 2017.

    Article  MathSciNet  Google Scholar 

  42. G. Gutin, M. Jones, B. Sheng, M. Wahlström, and A. Yeo. Chinese Postman Problem on edge-colored multigraphs. Discrete Appl. Math., 217, part 2:196–202, 2017.

    Article  MathSciNet  Google Scholar 

  43. G. Gutin, D. Karapetyan, and I. Razgon. Fixed-parameter algorithms in analysis of heuristics for extracting networks in linear programs. In IWPEC 2009, volume 5917 of Lect. Notes Comput. Sci., pages 222–233. Springer, 2009.

    Google Scholar 

  44. G. Gutin, E. J. Kim, S. Szeider, and A. Yeo. A probabilistic approach to problems parameterized above or below tight bounds. J. Comput. Syst. Sci., 77(2):422–429, 2011.

    Article  MathSciNet  Google Scholar 

  45. G. Gutin, I. Razgon, and E.J. Kim. Minimum leaf out-branching and related problems. Theor. Comput. Sci., 410:4571–4579, 2009.

    Article  MathSciNet  Google Scholar 

  46. G. Gutin and A. Yeo. On the number of connected convex subgraphs of a connected acyclic digraph. Discrete Appl. Math., 157(7):1660–1662, 2009.

    Article  MathSciNet  Google Scholar 

  47. F. Harary and E.M. Palmer. Graphical Enumeration. Academic Press, 1973.

    MATH  Google Scholar 

  48. A. Harutyunyan and B. Mohar. Planar digraphs of digirth five are 2-colorable. J. Graph Theory, 84(4):408–427, 2017.

    Article  MathSciNet  Google Scholar 

  49. F.S. Hillier and G.J. Lieberman. Introduction to Operations Research. McGraw Hill, 2001.

    Google Scholar 

  50. F.W. Horsley. Means Scheduling Manual: On-Time, On-Budget Construction Up-To-Date Computerized Scheduling. Roberts Means Co., 1991.

    Google Scholar 

  51. E.J. Kim and R. Williams. Improved parameterized algorithms for above average constraint satisfaction. In IPEC 2011, volume 7112 of Lect. Notes Comput. Sci., pages 118–131. Springer, 2012.

    Google Scholar 

  52. S. Kratsch, M. Pilipczuk, M. Pilipczuk, and M. Wahlström. Fixed-parameter tractability of multicut in directed acyclic graphs. SIAM J. Discrete Math., 29(1):122–144, 2015.

    Article  MathSciNet  Google Scholar 

  53. J. Kuipers and G. Moffa. Uniform random generation of large acyclic digraphs. Stat. Comput., 25:227–242, 2015.

    Article  MathSciNet  Google Scholar 

  54. F. Le Gall. Powers of tensors and fast matrix multiplication. In ISSAC 2014: International Symposium on Symbolic and Algebraic Computation, pages 296–303. ACM, 2014.

    Google Scholar 

  55. J. Leskovec, L. Backstrom, and J. Kleinberg. Meme-tracking and the dynamics of the news cycle. In KDD 2009: 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 497–506, 2009.

    Google Scholar 

  56. H. Li and G. Wang. Color degree and alternating cycles in edge-colored graphs. Discrete Math., 309:4349–4354, 2009.

    Article  MathSciNet  Google Scholar 

  57. V.A. Liskovets. On the number of maximal vertices of a random acyclic digraph. Theory Prob. Appl., 20(2):401–409, 1976.

    Article  MathSciNet  Google Scholar 

  58. A. Lo. A Dirac type condition for properly colored paths and cycles. J. Graph Theory, 76:60–87, 2014.

    Article  MathSciNet  Google Scholar 

  59. A. Lo. An edge-colored version of Dirac’s theorem. SIAM J. Discrete Math., 28:18–36, 2014.

    Article  MathSciNet  Google Scholar 

  60. A. Lo. Properly coloured Hamiltonian cycles in edge-coloured complete graphs. Combinatorica, 36:471–492, 2016.

    Article  MathSciNet  Google Scholar 

  61. S. Lohr. Study measures the chatter of the news cycle. New York Times, July 13th:B1, 2009. New York edition.

    Google Scholar 

  62. M. Mahajan, V. Raman, and S. Sikdar. Parameterizing above or below guaranteed values. J. Comput. Syst. Sci., 75(2):137–153, 2009.

    Article  MathSciNet  Google Scholar 

  63. B.D. McKay, F.E. Foggier, G.F. Royle, N.J.A. Sloane, I.M. Wanless, and H.S. Wilf. Acyclic digraphs and eigenvalues of (0, 1)-matrices. J. Integer Sequences, 7, 2004. article 04.3.3.

    Google Scholar 

  64. K. Mehlhorn. Data Structures and Algorithms 2: Graph Algorithms and NP-completeness. Springer-Verlag, Berlin, 1984.

    Book  Google Scholar 

  65. M. Mnich, G. Philip, S. Saurabh, and O. Suchý. Beyond max-cut: \(\lambda \)-extendible properties parameterized above the Poljak-Turzík bound. J. Comput. Syst. Sci., 80(7):1384–1403, 2014.

    Article  Google Scholar 

  66. K.G. Murty. Network programming. Prentice Hall, Englewood Cliffs, NJ, 1992.

    Google Scholar 

  67. V. Neumann-Lara. Vertex colorings in digraphs. some problems., 1985. Seminar notes, University of Waterloo.

    Google Scholar 

  68. Y. Perl and Y. Shiloach. Finding two disjoint paths between two pairs of vertices in a graph. J. Assoc. Comput. Mach., 25:1–9, 1978.

    Article  MathSciNet  Google Scholar 

  69. M. Pilipczuk and M. Wahlström. Directed multicut is W[1]-hard, even for four terminal pairs. In SODA 2016, pages 1167–1178. SIAM, 2016.

    Google Scholar 

  70. S. Poljak and D. Turzík. A polynomial time heuristic for certain subgraph optimization problems with guaranteed worst case bound. Discrete Math., 58(1):99–104, 1986.

    Article  MathSciNet  Google Scholar 

  71. L. Pozzi, K. Atasu, and P. Ienne. Exact and approximate algorithms for the extension of embedded processor instruction sets. IEEE Trans. Computer-Aided Design Integr. Circuits Syst., 25:1209–1229, 2006.

    Article  Google Scholar 

  72. V. Raman and S. Saurabh. Parameterized complexity of feedback set problems and their duals in tournaments. Theor. Comput. Sci., 351:446–458, 2006.

    Article  Google Scholar 

  73. J. Reddington. Improvements to instruction identification for custom instruction set design. PhD thesis, Royal Holloway, University of London, 2008.

    Google Scholar 

  74. J. Reddington, G. Gutin, A. Johnstone, E. Scott, and A. Yeo. Better than optimal: Fast identification of custom instruction candidates. In CSE 2009: 12th IEEE International Conference on Computational Science and Engineering, pages 17–24, 2009.

    Google Scholar 

  75. R.W. Robinson. Counting labeled acyclic digraphs. In F. Harary, editor, New Directions in the Theory of Graphs, pages 239–273. Acad. Press, 1973.

    Google Scholar 

  76. R.W. Robinson. Counting unlabeled acyclic digraphs. In C.H.C. Little, editor, Combinatorial Mathematics V: 5th Australian Conference, 1976, pages 28–43. Springer, 1977.

    Google Scholar 

  77. R.W. Robinson. Enumeration of acyclic digraphs. In The 2nd Chapel Hill Conf. on Combinatorial Mathematics and its Applications, pages 391–399, 1970.

    Google Scholar 

  78. N. Schwartges, J. Spoerhase, and A. Wolff. Approximation algorithms for the maximum leaf spanning tree problem on acyclic digraphs. In WAOA 2011, volume 7164 of Lect. Notes Comput. Sci., pages 77–88. Springer, 2012.

    Google Scholar 

  79. K. Simon. An improved algorithm for transitive closure on acyclic digraphs. Theor. Comput. Sci., 58:325–346, 1988.

    Article  MathSciNet  Google Scholar 

  80. J. Singer. A theorem in finite projective geometry and some applications to number theory. Trans. Am. Math. Soc., 43:377–385, 1938.

    Article  MathSciNet  Google Scholar 

  81. A. Slivkins. Parameterized tractability of edge-disjoint paths on directed acyclic graphs. SIAM J. Discrete Math., 24(1):146–157, 2010.

    Article  MathSciNet  Google Scholar 

  82. R.P. Stanley. Acyclic orientations of graphs. Discrete Math., 5:171–178, 1973.

    Article  MathSciNet  Google Scholar 

  83. R.P. Stanley. A matrix for counting paths in acyclic digraphs. J. Combin. Th. Ser. A, 74:169–172, 1996.

    Article  MathSciNet  Google Scholar 

  84. C. Thomassen. Paths, circuits and subdivisions. In Selected topics in graph theory Vol. 3, pages 97–131. Academic Press, 1988.

    Google Scholar 

  85. R. van Bevern, R. Bredereck, M. Chopin, S. Hartung, F. Hüffner, A. Nichterlein, and O. Suchý. Fixed-parameter algorithms for DAG partitioning. CoRR, abs/1611.08809, 2016.

    Google Scholar 

  86. R. van Bevern, R. Bredereck, M. Chopin, S. Hartung, F. Hüffner, A. Nichterlein, and O. Suchý. Parameterized complexity of DAG partitioning. In CIAC 2013, volume 7878 of Lect. Notes Comput. Sci., pages 49–60. Springer, 2013.

    Google Scholar 

  87. D.R. Wood. Bounded degree acyclic decompositions of digraphs. J. Combin. Theory, 90:309–313, 2004.

    Article  MathSciNet  Google Scholar 

  88. A. Yeo. A note on alternating cycles in edge-coloured graphs. J. Combin. Theory Ser. B, 69:222–225, 1997.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Royal Holloway University, Egham Hill, TW20 0EX, Egham, England

    Gregory Gutin

Authors
  1. Gregory Gutin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gregory Gutin .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark

    Jørgen Bang-Jensen

  2. Department of Computer Science, Royal Holloway, University of London, Egham, United Kingdom

    Gregory Gutin

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Gutin, G. (2018). Acyclic Digraphs. In: Bang-Jensen, J., Gutin, G. (eds) Classes of Directed Graphs. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-71840-8_3

Download citation

Publish with us

Policies and ethics

Search

Navigation