Graphs and Algorithms Solutions Exercise 1 (large Bipartite Subgraph)

We will create the partitioning by adding the vertices of V to A or B one by one. Let us assume V ′ is the set of vertices we have already added and at least half of the edges in G[V ′] are in between A and B. Now if u is the next vertex to insert we will greedily add it to A if |Γ(u) ∩ B| > |Γ(u) ∩ A| otherwise we add it to B. Thus after inserting u the invariant that at least half of the edges in G[V ′ ∪ {u}] still holds so adding the vertices greedily using this method will result in at least half of the edges being between A and B when we have inserted all of V .