The fitness landscape of the graph bipartitioning problem is investigated by performing a search space analysis for several types of graphs. The analysis shows that the structure of the search space is significantly different for the types of instances studied. Moreover, with increasing epistasis, the amount of gene interactions in the representation of a solution in an evolutionary algorithm, the number of local minima for one type of instance decreases and, thus, the search becomes easier. We suggest that other characteristics besides high epistasis might have greater influence on the hardness of a problem. To understand these characteristics, the notion of a dependency graph describing gene interactions is introduced. In particular, the local structure and the regularity of the dependency graph seems to be important for the performance of an algorithm, and in fact, algorithms that exploit these properties perform significantly better than others which do not. It will be shown that a simple hybrid multi-start local search exploiting locality in the structure of the graphs is able to find optimum or near optimum solutions very quickly. However, if the problem size increases or the graphs become unstructured, a memetic algorithm (a genetic algorithm incorporating local search) is shown to be much more effective.