The five basic classes of perfect graphs that form the base case of this structural decomposition are the bipartite graphs, line graphs of bipartite graphs, complementary graphs of bipartite graphs, complements of line graphs of bipartite graphs, and double split graphs. It is easy to see that bipartite graphs are perfect: in any nontrivial induced subgraph, the clique number and chromatic number are both two and therefore both equal. The perfection of complements of bipartite graphs, and of complements of line graphs of bipartite graphs, are both equivalent to Kőnig's theorem relating the sizes of maximum matchings, maximum independent sets, and minimum vertex covers in bipartite graphs. The perfection of line graphs of bipartite graphs can be stated equivalently as the fact that bipartite graphs have chromatic index equal to their maximum degree, proven by . Thus, all four of these basic classes are perfect. The double split graphs are a relative of the split graphs that can also be shown to be perfect.