Graph classes between parity and distance-hereditary graphs

Several graph problems (e.g., steiner tree, connected domination, hamiltonian path, and isomorphism problem), which can be solved in polynomial time for distance-hereditary graphs, are NP-complete or open for parity graphs. Moreover, the metric characterizations of these two graph classes suggest an excessive gap between them. We introduce a family of classes forming an infinite lattice with respect to inclusion, whose bottom and top elements are the class of the distance-hereditary graphs and the class of the parity graphs, respectively. We propose this family as a reference framework for studying the computational complexity of fundamental graph problems. To this purpose we characterize these classes using Cunningham decomposition and then use the devised structural characterization in order to show efficient algorithms for the recognition and isomorphism problems. As far as the isomorphism graph problem is concerned we find efficient algorithms for an infinite number of different classes (forming a chain with respect to the inclusion relation) in the family