Bounding the clique-width of H-free split graphs

A graph is H-free if it has no induced subgraph isomorphic to  H. We continue a study into the boundedness of clique-width of subclasses of perfect graphs. We identify five new classes of H-free split graphs whose clique-width is bounded. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs  H for which the class of H-free split graphs has bounded clique-width.