Convex Embeddings and Bisections of 3-Connected Graphs1

Given two disjoint subsets T 1 and T 2 of nodes in an undirected 3-connected graph G = (V, E) with node set V and arc set E, where \( {\left| {T_{1} } \right|}{\kern 1pt} \) and \( {\left| {T_{2} } \right|}{\kern 1pt} \) 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 \( {\left| {V_{1} \cap T_{j} } \right|} = {\left| {V_{2} \cap T_{j} } \right|} = {\left| {T_{j} } \right|}/2 \) holds for each j = 1,2. Such a partition can be found in \( O{\left( {{\left| V \right|}^{2} {\kern 1pt} \log {\kern 1pt} {\left| V \right|}} \right)} \) 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 an embedding always exists.

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Correspondence to Hiroshi Nagamochi.

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1 A preliminary version of this paper with title “Bisecting Two Subsets in 3-Connected Graphs” appeared in the Proceedings of the 10th Annual International Symposium on Algorithms and Computation, ISAAC ’99, (A. Aggarwal, C. P. Rangan, eds.), Springer LNCS 1741, 425–434, 1999.

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Nagamochi, H., Jordán, T., Nakao, Y. et al. Convex Embeddings and Bisections of 3-Connected Graphs1 . Combinatorica 22, 537–554 (2002).

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AMS Subject Classification (2000):

  • 05C40
  • 05C35