# Bisecting Two Subsets in 3-Connected Graphs

Conference paper

First Online:

## 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.

## Keywords

Undirected Graph Complete Bipartite Graph Convex Polytope Connected Subgraph Internal Edge
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.L. Chi-Yuan, J. Matoušek and W. Steiger,
*Algorithms for ham-sandwich cuts*, Discrete Comput. Geom., 11, 1994, 433–452.zbMATHGoogle Scholar - 2.M. E. Dyer and A. M. Frieze,
*On the complexity of partitioning graphs into connected subgraphs*, Discrete Applied Mathematics, 10, 1985, 139–153.zbMATHMathSciNetGoogle Scholar - 3.H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer-Verlag, Berlin, 1987.zbMATHGoogle Scholar
- 4.H. Edelsbrunner and R. Waupotitsch,
*Computing a ham-sandwich cut in two dimensions*, J. Symbolic Computation, 2, 1986, 171–178.zbMATHMathSciNetGoogle Scholar - 5.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.zbMATHMathSciNetGoogle Scholar - 6.E. Gyõri,
*On division of connected subgraphs*, Combinatorics (Proc. Fifth Hungarian Combinatorial Coll, 1976, Keszthely), Bolyai-North-Holland, 1978, 485–494.Google Scholar - 7.N. Linial, L. Lovász and A. Wigderson,
*Rubber bands, convex embeddings and graph connectivity*, Combinatorica, 8, 1988, 91–102.zbMATHGoogle Scholar - 8.L. Lovász,
*A homology theory for spanning trees of a graph*, Acta Math. Acad. Sci. Hungar, 30, 1977, 241–251.Google Scholar - 9.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.MathSciNetzbMATHGoogle Scholar - 10.E. Steinitz,
*Polyeder und Raumeinteilungen*, Encyklopädie der mathematischen Wissenschaften, Band III, Teil 1, 2. Hälfte, IIIAB12, 1916, 1–139.Google Scholar - 11.H. Suzuki, N. Takahashi and T. Nishizeki,
*A linear algorithm for bipartition of biconnected graphs*, Information Processing Letters, 33, 1990, 227–232.zbMATHMathSciNetGoogle Scholar - 12.W.T. Tutte,
*Connectivity in Graphs*, University of Toronto Press, 1966.Google Scholar - 13.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.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1999