Bisecting Two Subsets in 3-Connected Graphs
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.
KeywordsUndirected Graph Complete Bipartite Graph Convex Polytope Connected Subgraph Internal Edge
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