Partitioning graphs into equally large groups of nodes, minimizing the number of edges between different groups, is an extremely important problem in parallel computing. This paper presents genetic algorithms for suboptimal graph partitioning, with new crossover operators (KNUX, DKNUX) that lead to orders of magnitude improvement over traditional genetic operators in solution quality and speed. Our method can improve on good solutions previously obtained by using other algorithms or graph theoretic heuristics in, minimizing the total communication cost or the worst case cost of communication for a single processor. We also extend our algorithm to incremental graph partitioning problems, in which the graph structure or system properties changes with time. >