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Basic Terminology, Notation and Results

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Classes of Directed Graphs

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Abstract

In this chapter we will provide most of the terminology and notation used in this book. Various examples, figures and results should help the reader to better understand the notions introduced in the chapter. We also prove some basic results on digraphs and provide some fundamental digraph results without proofs. Most of our terminology and notation is standard and agrees with (Bang-Jensen, Gutin, Digraphs: theory, algorithms and applications. Springer, London, 2009, [4]).

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Notes

  1. 1.

    If we know from the context that D is directed, D may be called a graph.

  2. 2.

    By Menger’s theorem (Theorem 1.5.3), (1.1) is equivalent to the existence of k arc-disjoint dipaths from z to every other vertex of D.

  3. 3.

    The symmetric TSP is the problem of finding a minimum weight Hamilton cycle in a weighted complete undirected graph.

References

  1. 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 

  2. N. Alon, G. Gutin, E.J. Kim, S. Szeider, and A. Yeo. Solving MAX-\(r\)-SAT above a tight lower bound. Algorithmica, 61(3):638–655, 2011.

    Article  MathSciNet  Google Scholar 

  3. G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, and M. Protasi. Complexity and Approximation. Springer-Verlag, Berlin, 1999.

    Book  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  6. J. Bang-Jensen, J. Huang, and E. Prisner. In-tournament digraphs. J. Combin. Theory Ser. B, 59(2):267–287, 1993.

    Google Scholar 

  7. J. Bang-Jensen and A. Yeo. The minimum spanning strong subdigraph problem is fixed parameter tractable. Discrete Appl. Math., 156:2924–2929, 2008.

    Article  MathSciNet  Google Scholar 

  8. M. Basavaraju, P. Misra, M.S. Ramanujan, and S. Saurabh. On finding highly connected spanning subgraphs. CoRR, arXiv:1701.02853, 2017.

  9. I. Bezáková, R. Curticapean, H. Dell, and F.V. Fomin. Finding detours is fixed-parameter tractable. In ICALP 2017, volume 80 of LIPIcs, pages 54:1–54:14, 2017.

    Google Scholar 

  10. D. Binkele-Raible, H. Fernau, F.V. Fomin, D. Lokshtanov, S. Saurabh, and Y. Villanger. Kernel(s) for problems with no kernel: On out-trees with many leaves. ACM Trans. Algorithms, 8(4):38:1–38:19, 2012.

    Article  Google Scholar 

  11. H.L. Bodlaender, R.G. Downey, M.R. Fellows, and D. Hermelin. On problems without polynomial kernels. J. Comput. Syst. Sci., 75(8):423–434, 2009.

    Article  MathSciNet  Google Scholar 

  12. F. Boesch and R. Tindell. Robbins’s theorem for mixed multigraphs. Amer. Math. Mon., 87(9):716–719, 1980.

    Article  MathSciNet  Google Scholar 

  13. N. Christofides. Worst-case analysis of a new heuristic for the traveling salesman problem. Technical Report CS-93-13, Carnegie Mellon University, 1976.

    Google Scholar 

  14. M. Cygan, F.V. Fomin, A. Golovnev, A.S. Kulikov, I. Mihajlin, J. Pachocki, and A. Socala. Tight bounds for graph homomorphism and subgraph isomorphism. In SODA 2016: 27th ACM-SIAM Symposium on Discrete Algorithms, pages 1643–1649, 2016.

    Google Scholar 

  15. M. Cygan, F.V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer, 2015.

    Chapter  Google Scholar 

  16. J. Daligault, G. Gutin, E.J. Kim, and A. Yeo. FPT algorithms and kernels for the directed \(k\)-leaf problem. J. Comput. Syst. Sci., 76:144–152, 2010.

    Article  MathSciNet  Google Scholar 

  17. 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 

  18. R.G. Downey and M.R. Fellows. Fundamentals of Parameterized Complexity. Springer, 2013.

    Google Scholar 

  19. F.V. Fomin and D. Kratsch. Exact Exponential Algorithms. Springer, 2010.

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. T. Gallai and A.N. Milgram. Verallgemeinerung eines graphentheoretischen Satzes von Rédei. Acta Sci. Math. Szeged, 21:181–186, 1960.

    Google Scholar 

  22. G. Gutin. Characterization of complete \(n\)-partite digraphs that have a Hamiltonian path. Kibernetika (Kiev), no. 1:107–108, 1988.

    Google Scholar 

  23. G. Gutin and R. Li. Seymour’s second neighbourhood conjecture for quasi-transitive oriented graphs. CoRR, abs/1704.01389, 2017.

    Google Scholar 

  24. G. Gutin, M.S. Ramanujan, F. Reidl, and M. Wahlström. Path-contractions, edge deletions and connectivity preservation. In ESA 2017: 25th Annual European Symposium on Algorithms, volume 87 of LIPIcs, pages 47:1–47:13, 2017.

    Google Scholar 

  25. G. Gutin, M. Wahlström, and A. Yeo. Rural postman parameterized by the number of components of required edges. J. Comput. Syst. Sci., 83(1):121–131, 2017.

    Article  MathSciNet  Google Scholar 

  26. A.J. Hoffman. Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. In R. Bellman and M. Hall, editors, Combinatorial Analysis, pages 113–128. American Mathematical Society, Providence, RI, 1960.

    Chapter  Google Scholar 

  27. D. Kühn and D. Osthus. Linkedness and ordered cycles in digraphs. Combin. Prob. Comput., 17:411–422, 2008.

    Article  MathSciNet  Google Scholar 

  28. M. Las Vergnas. Sur les arborescences dans un graphe orienté. Discrete Math., 15(1):27–39, 1976.

    Article  MathSciNet  Google Scholar 

  29. K. Menger. Zur allgemeinen Kurventheorie. Fund. Math., 10:96–115, 1927.

    Article  Google Scholar 

  30. J.W. Moon. On subtournaments of a tournament. Can. Math. Bull., 9:297–301, 1966.

    Article  MathSciNet  Google Scholar 

  31. R.A. Moser and D. Scheder. A full derandomization of Schöning’s \(k\)-SAT algorithm. In STOC 2011: 43rd ACM Symposium on Theory of Computing, pages 245–252, 2011.

    Google Scholar 

  32. L. Rédei. Ein kombinatorischer Satz. Acta Litt. Szeged, 7:39–43, 1934.

    Google Scholar 

  33. H.E. Robbins. A theorem on graphs with an application to a problem on traffic control. Amer. Math. Mon., 46:281–283, 1939.

    Article  Google Scholar 

  34. U. Schöning. A probabilistic algorithm for \(k\)-SAT and constraint satisfaction problems. In FOCS 1999: 40th IEEE Symposium on Foundations of Computer Science, pages 410–414, 1999.

    Google Scholar 

  35. Y. Shiloach. Edge-disjoint branching in directed multigraphs. Inform. Process. Lett., 8(1):24–27, 1979.

    Article  MathSciNet  Google Scholar 

  36. M. Sorge. Some algorithmic challenges in arc routing, May 2013. Talk at NII Shonan Seminar no. 18.

    Google Scholar 

  37. M. Sorge, R. van Bevern, R. Niedermeier, and M. Weller. A new view on rural postman based on eulerian extension and matching. J. Discrete Algor., 16:12–33, 2012.

    Article  MathSciNet  Google Scholar 

  38. D.P. Williamson and D.B. Shmoys. The Design of Approximation Algorithms. Cambridge University Press, 2011.

    Google Scholar 

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Correspondence to Gregory Gutin .

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Bang-Jensen, J., Gutin, G. (2018). Basic Terminology, Notation and Results. 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_1

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