On a bipartition problem of Bollobás and Scott

The bipartite density of a graphG is max{|E(H)|/|E(G)| : H is a bipartite subgraph of G}. It is NP-hard to determine the bipartite density of any triangle-free cubic graph. A biased maximum bipartite subgraph of a graph G is a bipartite subgraph of G with the maximum number of edges such that one of its partite sets is independent in G. Let H denote the collection of all connected cubic graphs which have bipartite density 4 5 and contain biased maximum bipartite subgraphs. Bollobás and Scott asked which cubic graphs belong to H. This same problem was also proposed by Malle in 1982. We show that any graph in H can be reduced, through a sequence of three types of operations, to a member of a well characterized class. As a consequence, we give an algorithm that decides whether a given graph G belongs to H. Our algorithm runs in polynomial time, provided that G has a constant number of triangles that are not blocks of G and do not share edges with any other triangles in G.