Convex Embeddings and Bisections of 3-Connected Graphs 1

Abstract

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 V1 and V2 such that the graphs induced by V1 and V2 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 Rk-1 and prove that for k = 3 such an embedding always exists.