On the equivalence in complexity among basic problems on bipartite and parity graphs

Parity graphs form a superclass of bipartite graphs. Since their introduction, all the algorithms proposed as solutions to the recognition problem and other computational problems exploit the structural property described by Burlet and Uhry. This paper newly describes a different structural property based on split decomposition for parity graphs. This result, together with the observation that the split decomposition process can be performed in linear time, allows us to provide optimum algorithms for both the recognition problem and the maximum weighted clique problem in this class. We further propose a general algorithm to solve the maximum weighted independent set problem for certain classes of graphs. Its application to parity graphs provides the best solution for this problem. Moreover, when the algorithm is applied to distance-hereditary graphs, it equals the best known solution, which is also the optimum for this class. A remarkable consequence of this work is that the extension of bipartite graphs to parity graphs does not increase the complexity of these basic problems, since the worst case occurs when the parity graph is an undecomposable bipartite graph.