Abstract
We enumerate all of the shortest paths between any vertex v and the identity vertex in an (n, k)-star graph by enumerating the minimum factorizations of v in terms of the transpositions corresponding to edges in that graph. This result generalizes a previous one for the star graph, and can be applied to obtain the number of the shortest paths between a pair of vertices in some of the other similar structures. It also implies an algorithm to enumerate all such paths.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Akers, S.B., Krishnamurthy, B.: A Group Theoretic Model for Symmetric Interconnection Networks. IEEE Trans. on Computers 38(4), 555–566 (1989)
Cheng, E., Grossman, J., Lipták, L., Qiu, K., Shen, Z.: Distance Formula and Shortest Paths for the (n,k)-Star Graphs. Information Sciences 180, 1671–1680 (2010)
Chiang, W., Chen, R.: The (n, k)-Star graph: A Generalized Star Graph. Information Processing Letters 56, 259–264 (1995)
Day, K., Tripathi, A.: Arrangement Graphs: A Class of Generalized Star Graphs. Information Processing Letter 42, 235–241 (1992)
Denés, J.: The Representation of Permutation as the Product of a Minimal Number of Transpositions and its Connection with the Theory of Graphs. Magyar Tudományos Akadémia. Matematikai Kutatóintézet 4, 63–71 (1959)
Fàbrega, J., Fiol, M.A.: Maximally Connected Digraphs. J. Graph Theory 13(6), 657–668 (1989)
García, F., Solano, J., Stojmenović, I., Stojmenović, M.: Higher Dimensional Hexagonal Networks. J. Parallel and Distributed Computing 63, 1164–1172 (2003)
Goupil, A., Schaeffer, G.: Factoring N-Cycles and Counting Maps of Given Genus. Europ. J. Combinatorics 19, 819–834 (1998)
Gross, J., Yellen, J. (eds.): Handbook of Graph Theory. CRC Press, Boca Raton (2003)
Irving, J.: On the Number of Factorizations of a Full Cycle. J. Combinatorial Theory, Series A 113, 1549–1554 (2006)
Irving, J., Rattan, A.: Factorizations of Permutations into Star Transpositions. Discrete Mathematics 309(6), 1435–1442 (2009)
Latifi, S., Srimani, P.: Transposition Networks as a Class of Fault-Tolerant Robust Networks. IEEE Trans. on Computers 45(2), 230–238 (1996)
Marcote, X., Balbuena, C., Pelayo, I.: Diameter, Short Paths and Superconnectivity in Diagraphs. Discrete Mathematics 288, 113–123 (2004)
Pak, I.: Reduced Decompositions of Permutations in Terms of Star Transpositions, Generalized Catalan Numbers and k-ary Trees. Discrete Mathematics 204, 329–335 (1999)
Schwiebert, L.: There is no Optimal Routing Policy for the Torus. Information Processing Letters 83, 331–336 (2002)
Shen, Z., Qiu, K., Cheng, E.: On the Surface Area of the (n,k)-Star Graph. Theoretical Computer Science 410, 5481–5490 (2009)
Stanley, R.P.: On the Number of Reduced Decompositions of Elements of Coxeter Group. Europ. J. Combinatorics 5, 359–372 (1984)
Tang, J., Balbuena, C., Lin, Y., Miller, M.: An Open Problem: Superconnectivity of Regular Digraphs with Respect to Semigirth and Diameter. In: Program of International Workshop on Optimal Network Topologies (IWONT 2007), Pilsen-Černice, Czech Republic (September 2007), http://iti.zcu.cz/iwont2007/iwont2007.pdf
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
Cheng, E., Qiu, K., Shen, Z.Z. (2010). The Number of Shortest Paths in the (n, k)-Star Graphs. In: Wu, W., Daescu, O. (eds) Combinatorial Optimization and Applications. COCOA 2010. Lecture Notes in Computer Science, vol 6508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17458-2_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-17458-2_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17457-5
Online ISBN: 978-3-642-17458-2
eBook Packages: Computer ScienceComputer Science (R0)