Finding a maximum-weight induced kk-partite subgraph of an ii-triangulated graph

An ii-triangulated graph   is a graph in which every odd cycle has two non-crossing chords; ii-triangulated graphs form a subfamily of perfect graphs. A slightly more general family of perfect graphs are clique-separable   graphs. A graph is clique-separable precisely if every induced subgraph either has a clique cutset, or is a complete multipartite graph or a clique joined to an arbitrary bipartite graph. We exhibit a polynomial time algorithm for finding a maximum-weight induced kk-partite subgraph of an ii-triangulated graph, and show that the problem of finding a maximum-size bipartite induced subgraph in a clique-separable graph is NP-complete.