From the observation, that a bipartite graph doesn't contain odd cycles, it would seem natural to attempt to destroy all odd cycles in the most efficient way, by either removing edges or vertices of odd cycles, in order to find a maximal subset of the vertices that spans a bipartite sub graph.

I would appreciate references to articles in which such direct methods of destroying odd cycles by removing edges or vertices from a graph and their related problems are discussed.

An example of a problem that is related to most effectively destroying odd cycles, would be how to determine the number of odd cycles that removing an edge or vertex will destroy.