Abstract
An orientation of an undirected graph G is a directed graph D on V(G) with exactly one of directed edges (u, v) and (v, u) for each pair of vertices u and v adjacent in G. For integer k ≥ 3, we say a directed graph D is k-cyclic if every edge of D belongs to a directed cycle in D of length at most k. We consider the problem of deciding if a given graph has a k-cyclic orientation. We show that this problem is NP-complete for every fixed k ≥ 3 for general graphs and for every fixed k ≥ 4 for planar graphs. We give a polynomial time algorithm for planar graphs with k = 3, which constructs a 3-cyclic orientation when the answer is affirmative.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Chvátal, V., Thomassen, C.: Distances in orientations of graphs. Journal of Combinatorial Theory, Series B 24(1), 65–75 (1978)
Dankelmann, P., Oellermann, O.R., Wu, J.-L.: Minimum average distance of strong orientations of graphs. Discrete Appl. Math. 143(1-3), 204–212 (2004)
Eggemann, N., Noble, S.D.: Minimizing the oriented diameter of a planar graph. Electronic Notes in Discrete Mathematics 34(1), 267–271 (2009)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)
Louis Hakimi, S., Schmeichel, E.F., Young, N.E.: Orienting graphs to optimize reachability. Information Processing Letters 63, 229–235 (1997)
Ito, T., Miyamoto, Y., Ono, H., Tamaki, H., Uehara, R.: Route-enabling graph orientation problems. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 403–412. Springer, Heidelberg (2009)
Robbins, H.E.: A theorem on graphs, with an application to a problem of traffic control. The American Mathematical Monthly 46(5), 281–283 (1939)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kobayashi, Y., Miyamoto, Y., Tamaki, H. (2010). k-cyclic Orientations of Graphs. In: Cheong, O., Chwa, KY., Park, K. (eds) Algorithms and Computation. ISAAC 2010. Lecture Notes in Computer Science, vol 6507. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17514-5_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-17514-5_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17513-8
Online ISBN: 978-3-642-17514-5
eBook Packages: Computer ScienceComputer Science (R0)