Abstract
Given a directed graph D = (V,A) with a set of d specified vertices S = {s 1,…, s d } ⊆ V and a function f: S → ℕ where ℕ denotes the set of natural numbers, we present a necessary and sufficient condition such that there exist Σ d i=1 f(s i ) arc-disjoint in-trees denoted by T i,1,T i,2,…, \( T_{i,f(s_0 )} \) for every i = 1,…,d such that T i,1,…,\( T_{i,f(s_0 )} \) are rooted at s i and each T i,j spans the vertices from which s i is reachable. This generalizes the result of Edmonds [2], i.e., the necessary and sufficient condition that for a directed graph D=(V,A) with a specified vertex s∈V, there are k arc-disjoint in-trees rooted at s each of which spans V. Furthermore, we extend another characterization of packing in-trees of Edmonds [1] to the one in our case.
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J. Edmonds: Submodular functions, matroids, and certain polyhedra; in Combinatorial Structures and their Applications (R. Guy, H. Hanani, N. Sauer, and J. Schönheim, editors), pages 69–87. Gordon and Breach, New York, 1970.
J. Edmonds: Edge-disjoint branchings, in Combinatorial Algorithms (R. Rustin, editor), pages 91–96. Academic Press, New York, 1973.
H. N. Gabow: A matroid approach to finding edge connectivity and packing arborescences, Journal of Computer and System Sciences 50(2) (1995), 259–273.
N. Kamiyama, N. Katoh and A. Takizawa: An efficient algorithm for the evacuation problem in a certain class of a network with uniform path-lengths, in Proceedings of the third International Conference on Algorithmic Aspects in Information and Management, volume 4508 of Lecture Notes in Computer Science, pages 178–190. Springer, 2007.
L. Lovász: On two minimax theorems in graph, J. Comb. Theory, Ser. B 21(2) (1976), 96–103.
A. Schrijver: Combinatorial Optimization: Polyhedra and Efficiency; Springer, 2003.
P. Tong and E. L. Lawler: A faster algorithm for finding edge-disjoint branchings, Inf. Process. Lett. 17(2) (1983), 73–76.
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Supported by JSPS Research Fellowships for Young Scientists.
Supported by the project New Horizons in Computing, Grand-in-Aid for Scientific Research on Priority Areas, MEXT Japan.
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Kamiyama, N., Katoh, N. & Takizawa, A. Arc-disjoint in-trees in directed graphs. Combinatorica 29, 197–214 (2009). https://doi.org/10.1007/s00493-009-2428-z
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DOI: https://doi.org/10.1007/s00493-009-2428-z