Abstract
An edge-bisector of a graph is a set of edges whose removing separates the graph into two subgraphs of same order, within one. The edge-bisection of a graph is the cardinality of the smallest edge-bisector. The main purpose of this paper is to estimate the quality of general bounds on the edge-bisection of Cayley graphs. For this purpose we have focused on chordal rings of degree 3. These graphs are Cayley graphs on the dihedral group and can be considered as the simplest Cayley graphs on a non-abelian group (the dihedral group is metabelian). Moreover, the natural plane tessellation used to represent and manipulate these graphs can be generalized to other types of tessellations including abelian Cayley graphs. We have improved previous bounds on the edge-bisection of chordal rings and we have shown that, for any fixed chord, our upper bound on the edge-bisection of chordal rings is optimal up to an O(log n) factor. Finally, we have given tight bounds for optimal chordal rings, that are those with the maximum number of vertices for a given diameter.
Work supported by the Spanish Research Council (Comisión Interministerial de Ciencia y Tecnología, CICYT) under project TIC-97-0963.
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References
A. Agarwal. Limits on interconnection network performance. IEEE Transactions on Parallel and Distributed Systems, 2(4):398–412, October 1991.
F. Annexstein and M. Baumslag. On the diameter and bisector size of Cayley graphs. Math. Systems Theory, 26:271–291, 1993.
B. Arden and H. Lee. Analysis of chordal ring network. IEEE Trans. Comput., C-30(4):291–295, April 1981.
S. Arora, D. Karger, and M. Karpinski. Polynomial time approximation schemes for dense instances of NP-hard problems. In Proceedings of the 27th Annual ACM Symposium on Theory of Computing (STOC-95), pages 284–293. ACM Press, 1995.
L. Barriére. Triangulations and chordal rings. In 6th Int. Colloquium on Structural Information and Communication Complexity (SIROCCO), volume 5 of Proceedings in Informatics, pages 17–31. Carleton Scientific, 1999.
L. Barriére, J. Cohen, and M. Mitjana. Gossiping in chordal rings under the line model. In J. Hromkovic and W. Unger, editors, Proceedings of the MFCS’98 Workshop on Communication, pages 37–47, 1998.
S. Blackburn. Node bisectors of Cayley graphs. Mathematical Systems Theory, 29:589–598, 1996.
R. Boppana. Eigenvalues and graph bisection: An average-case analysis. In Ashok K. Chandra, editor, Proceedings of the 28th Annual Symposium on Foundations of Computer Science, pages 280–285, Los Angeles, CA, October 1987. IEEE Computer Society Press.
T. Bui, S. Chaudhuri, T. Leighton, and M. Sipser. Graph bisection algorithms with good average case behavior. In 25th Annual Symposium on Foundations of Computer Science, pages 181–192, Los Angeles, Ca., USA, October 1984. IEEE Computer Society Press.
D. Dolev, J. Halpern, B. Simons, and R. Strong. A new look at fault tolerant network routing. In Proc. ACM 16th STOC, pages 526–535, 1984.
W. Donath and A. Hoffman. Lower bounds for the partitioning of graphs. IBM Journal of Research and Developement, 17:420–425, 1973.
S. Even, A. Itai, and A. Shamir. On the complexity of timetable and multicommodity flow problems. SIAM Journal of Computing, 5(4):691–703, 1976.
N. Garey and D. Johnson. Computers and Intractability-A Guide to Theory of NP-completeness. W.H. Freeman and Company, 1979.
M.-C. Heydemann, J.-C. Meyer, J. Opatrny, and D. Sotteau. Forwarding indices of consistent routings and their complexity. Networks, 24:75–82, 1994.
J. Hromkovic, R. Klasing, W. Unger, and H. Wagener. Optimal algorithms for broadcast and gossip in edge-disjoint path modes. In (SWAT’94), Lecture Notes in Computer Science 824, 1994.
B. Kerninghan and S. Lin. An efficient heuristic procedure for partitioning graphs. Bell Systems Technology Journal, 49:291–307, 1970.
F.T. Leighton. Introduction to Parallel Algorithms and Architectures: Arrays, Trees and Hypercubes. Morgan Kaufmann Publishers, 1992.
R. Lipton and R. Tarjan. A separator theorem for planar graphs. In WATERLOO: Proceedings of a Conference on Theoretical Computer Science, 1977.
Jerrum M and G. Sorkin. Simulated annealing for graph bisection. In 34th Annual Symposium on Foundations of Computer Science, pages 94–103, Palo Alto, California, 3–5 November 1993. IEEE.
P. Morillo. Grafos y digrafos asociados con teselaciones como modelos para redes de interconexión. PhD thesis, Universitat Politécnica de Catalunya, Barcelona, Spain, 1987.
D. Robinson. A Course in Theory of Groups, volume 80 of Graduate Texts in Math. Springer, 1996.
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Barriére, L., Fábrega, J. (2000). Edge-Bisection of Chordal Rings. In: Nielsen, M., Rovan, B. (eds) Mathematical Foundations of Computer Science 2000. MFCS 2000. Lecture Notes in Computer Science, vol 1893. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44612-5_12
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DOI: https://doi.org/10.1007/3-540-44612-5_12
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