Arborescence (graph theory)

In graph theory, an arborescence is a directed graph in which, for a vertex u called the root and any other vertex v, there is exactly one directed path from u to v. An arborescence is thus the directed-graph form of a rooted tree, understood here as an undirected graph.[1][2]

Equivalently, an arborescence is a directed, rooted tree in which all edges point away from the root; a number of other equivalent characterizations exist.[3][4] Every arborescence is a directed acyclic graph (DAG), but not every DAG is an arborescence.

An arborescence can equivalently be defined as a rooted digraph in which the path from the root to any other vertex is unique.[1]

The term arborescence comes from French.[5] Some authors object to it on grounds that it is cumbersome to spell.[6] There is a large number of synonyms for arborescence in graph theory, including directed rooted tree[2][6] out-arborescence,[7] out-tree,[8] and even branching being used to denote the same concept.[8] Rooted tree itself has been defined by some authors as a directed graph.[9][10][11]

Furthermore, some authors define an arborescence to be a spanning directed tree of a given digraph.[11][12] The same can be said about some of its synonyms, especially branching.[12] Other authors use branching to denote a forest of arborescences, with the latter notion defined in broader sense given at beginning of this article,[13][14] but a variation with both notions of the spanning flavor is also encountered.[11]

It's also possible to define a useful notion by reversing all the arcs of an arborescence, i.e. making them all point to the root rather than away from it. Such digraphs are also designated by a variety of terms such as in-tree[15] or anti-arborescence[16] etc. W. T. Tutte distinguishes between the two cases by using the phrases arborescence diverging from [some root] and arborescence converging to [some root].[17]

The number of rooted trees (or arborescences) with n nodes is given by the sequence:

0, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486, ... (sequence A000081 in the OEIS).

See also

References
  1. Gordon, Gary (1989). "A greedoid polynomial which distinguishes rooted arborescences". Proceedings of the American Mathematical Society. 107 (2): 287. doi:10.1090/S0002-9939-1989-0967486-0.
  2. Stanley Gill Williamson (1985). Combinatorics for Computer Science. Courier Dover Publications. p. 288. ISBN 978-0-486-42076-9.
  3. Jean-Claude Fournier (2013). Graphs Theory and Applications: With Exercises and Problems. John Wiley & Sons. pp. 94–95. ISBN 978-1-84821-070-7.
  4. Jean Gallier (2011). Discrete Mathematics. Springer Science & Business Media. pp. 193–194. ISBN 978-1-4419-8046-5.
  5. Per Hage and Frank Harary (1996). Island Networks: Communication, Kinship, and Classification Structures in Oceania. Cambridge University Press. p. 43. ISBN 978-0-521-55232-5.
  6. Mehran Mesbahi; Magnus Egerstedt (2010). Graph Theoretic Methods in Multiagent Networks. Princeton University Press. p. 38. ISBN 1-4008-3535-6.
  7. Ding-Zhu Du; Ker-I Ko; Xiaodong Hu (2011). Design and Analysis of Approximation Algorithms. Springer Science & Business Media. p. 108. ISBN 978-1-4614-1701-9.
  8. Jonathan L. Gross; Jay Yellen; Ping Zhang (2013). Handbook of Graph Theory, Second Edition. CRC Press. p. 116. ISBN 978-1-4398-8018-0.
  9. David Makinson (2012). Sets, Logic and Maths for Computing. Springer Science & Business Media. pp. 167–168. ISBN 978-1-4471-2499-3.
  10. Kenneth Rosen (2011). Discrete Mathematics and Its Applications, 7th edition. McGraw-Hill Science. p. 747. ISBN 978-0-07-338309-5.
  11. Alexander Schrijver (2003). Combinatorial Optimization: Polyhedra and Efficiency. Springer. p. 34. ISBN 3-540-44389-4.
  12. Jørgen Bang-Jensen; Gregory Z. Gutin (2008). Digraphs: Theory, Algorithms and Applications. Springer. p. 339. ISBN 978-1-84800-998-1.
  13. James Evans (1992). Optimization Algorithms for Networks and Graphs, Second Edition. CRC Press. p. 59. ISBN 978-0-8247-8602-1.
  14. Bernhard Korte; Jens Vygen (2012). Combinatorial Optimization: Theory and Algorithms (5th ed.). Springer Science & Business Media. p. 18. ISBN 978-3-642-24488-9.
  15. Kurt Mehlhorn; Peter Sanders (2008). Algorithms and Data Structures: The Basic Toolbox (PDF). Springer Science & Business Media. p. 52. ISBN 978-3-540-77978-0.
  16. Bernhard Korte; Jens Vygen (2012). Combinatorial Optimization: Theory and Algorithms (5th ed.). Springer Science & Business Media. p. 28. ISBN 978-3-642-24488-9.
  17. Tutte, W.T. (2001), Graph Theory, Cambridge University Press, pp. 126–127, ISBN 978-0-521-79489-3


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