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]).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
If we know from the context that D is directed, D may be called a graph.
- 2.
- 3.
The symmetric TSP is the problem of finding a minimum weight Hamilton cycle in a weighted complete undirected graph.
References
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.
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.
G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, and M. Protasi. Complexity and Approximation. Springer-Verlag, Berlin, 1999.
J. Bang-Jensen and G. Gutin. Digraphs: Theory, Algorithms and Applications. Springer-Verlag, London, 2nd edition, 2009.
J. Bang-Jensen and G. Gutin. Out-branchings with extremal number of leaves. Ramanujan Math. Soc. Lect. Notes, 13:91–99, 2010.
J. Bang-Jensen, J. Huang, and E. Prisner. In-tournament digraphs. J. Combin. Theory Ser. B, 59(2):267–287, 1993.
J. Bang-Jensen and A. Yeo. The minimum spanning strong subdigraph problem is fixed parameter tractable. Discrete Appl. Math., 156:2924–2929, 2008.
M. Basavaraju, P. Misra, M.S. Ramanujan, and S. Saurabh. On finding highly connected spanning subgraphs. CoRR, arXiv:1701.02853, 2017.
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.
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.
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.
F. Boesch and R. Tindell. Robbins’s theorem for mixed multigraphs. Amer. Math. Mon., 87(9):716–719, 1980.
N. Christofides. Worst-case analysis of a new heuristic for the traveling salesman problem. Technical Report CS-93-13, Carnegie Mellon University, 1976.
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.
M. Cygan, F.V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer, 2015.
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.
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.
R.G. Downey and M.R. Fellows. Fundamentals of Parameterized Complexity. Springer, 2013.
F.V. Fomin and D. Kratsch. Exact Exponential Algorithms. Springer, 2010.
S. Fortune, J.E. Hopcroft, and J. Wyllie. The directed subgraph homeomorphism problem. Theor. Comput. Sci., 10:111–121, 1980.
T. Gallai and A.N. Milgram. Verallgemeinerung eines graphentheoretischen Satzes von Rédei. Acta Sci. Math. Szeged, 21:181–186, 1960.
G. Gutin. Characterization of complete \(n\)-partite digraphs that have a Hamiltonian path. Kibernetika (Kiev), no. 1:107–108, 1988.
G. Gutin and R. Li. Seymour’s second neighbourhood conjecture for quasi-transitive oriented graphs. CoRR, abs/1704.01389, 2017.
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.
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.
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.
D. Kühn and D. Osthus. Linkedness and ordered cycles in digraphs. Combin. Prob. Comput., 17:411–422, 2008.
M. Las Vergnas. Sur les arborescences dans un graphe orienté. Discrete Math., 15(1):27–39, 1976.
K. Menger. Zur allgemeinen Kurventheorie. Fund. Math., 10:96–115, 1927.
J.W. Moon. On subtournaments of a tournament. Can. Math. Bull., 9:297–301, 1966.
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.
L. Rédei. Ein kombinatorischer Satz. Acta Litt. Szeged, 7:39–43, 1934.
H.E. Robbins. A theorem on graphs with an application to a problem on traffic control. Amer. Math. Mon., 46:281–283, 1939.
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.
Y. Shiloach. Edge-disjoint branching in directed multigraphs. Inform. Process. Lett., 8(1):24–27, 1979.
M. Sorge. Some algorithmic challenges in arc routing, May 2013. Talk at NII Shonan Seminar no. 18.
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.
D.P. Williamson and D.B. Shmoys. The Design of Approximation Algorithms. Cambridge University Press, 2011.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-71840-8_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-71839-2
Online ISBN: 978-3-319-71840-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)