Bisecting two subsets in 3-connected graphs

Abstract

Given two subsets T1 and T2 of vertices in a 3-connected graph G = (V, E), where |T1| and |T2| 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 |V1 ∩ Tj|= |V2 ∩ Tj| = |Tj |/2 holds for each j = 1, 2. Such a partition can be found in 0(|V|2) time. Our proof relies on geometric arguments. We define a new type of 'convex embedding' of к-connected graphs into real space Rk_1 and prove that for к = 3 such embedding always exists.