Abstract
Given two subsets T 1 and T 2 of vertices in a 3-connected graph G = (V,E), where |T 1| and |T 2| are even numbers, we show that V can be partitioned into two sets V 1 and V 2 such that the graphs induced by V 1 and V 2 are both connected and |V 1∩T j| = |V 2∩T j| = |T j|/2 holds for each j = 1, 2. Such a partition can be found in O(|V|2) time. Our proof relies on geometric arguments. We define a new type of ‘convex embedding’ of k-connected graphs into real space R k-1 and prove that for k = 3 such embedding always exists.
This research was partially supported by the Scientic Grant-in-Aid from Ministry of Education, Science, Sports and Culture of Japan, and the subsidy from the In- amori Foundation. Part of this work was done while the second author visited the Department of Applied Mathematics and Physics at Kyoto University, supported by the Monbusho International Scientic Research Program no. 09044160.
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
L. Chi-Yuan, J. Matoušek and W. Steiger, Algorithms for ham-sandwich cuts, Discrete Comput. Geom., 11, 1994, 433–452.
M. E. Dyer and A. M. Frieze, On the complexity of partitioning graphs into connected subgraphs, Discrete Applied Mathematics, 10, 1985, 139–153.
H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer-Verlag, Berlin, 1987.
H. Edelsbrunner and R. Waupotitsch, Computing a ham-sandwich cut in two dimensions, J. Symbolic Computation, 2, 1986, 171–178.
M. R. Garey, D. S. Johnson and R. E. Tarjan, The planar Hamiltonian circuit problem is NP-complete, SIAM J. Comput., 5, 1976, 704–714.
E. Gyõri, On division of connected subgraphs, Combinatorics (Proc. Fifth Hungarian Combinatorial Coll, 1976, Keszthely), Bolyai-North-Holland, 1978, 485–494.
N. Linial, L. Lovász and A. Wigderson, Rubber bands, convex embeddings and graph connectivity, Combinatorica, 8, 1988, 91–102.
L. Lovász, A homology theory for spanning trees of a graph, Acta Math. Acad. Sci. Hungar, 30, 1977, 241–251.
H. Nagamochi and T. Ibaraki, A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph, Algorithmica, 7, 1992, 583–596.
E. Steinitz, Polyeder und Raumeinteilungen, Encyklopädie der mathematischen Wissenschaften, Band III, Teil 1, 2. Hälfte, IIIAB12, 1916, 1–139.
H. Suzuki, N. Takahashi and T. Nishizeki, A linear algorithm for bipartition of biconnected graphs, Information Processing Letters, 33, 1990, 227–232.
W.T. Tutte, Connectivity in Graphs, University of Toronto Press, 1966.
K. Wada and K. Kawaguchi, Efficient algorithms for tripartitioning triconnected graphs and 3-edge-connected graphs, Lecture Notes in Comput. Sci., 790, Springer, Graph-theoretic concepts in computer science, 1994, 132–143.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Nagamochi, H., Nakao, Y., Ibaraki, T., Jordán, T. (1999). Bisecting Two Subsets in 3-Connected Graphs. In: Algorithms and Computation. ISAAC 1999. Lecture Notes in Computer Science, vol 1741. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46632-0_43
Download citation
DOI: https://doi.org/10.1007/3-540-46632-0_43
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66916-6
Online ISBN: 978-3-540-46632-1
eBook Packages: Springer Book Archive