Split Permutation Graphs

The class of split permutation graphs is the intersection of two important classes, the split graphs and permutation graphs. It also contains an important subclass, the threshold graphs. The class of threshold graphs enjoys many nice properties. In particular, these graphs have bounded clique-width and they are well-quasi-ordered by the induced subgraph relation. It is known that neither of these two properties is extendable to split graphs or to permutation graphs. In the present paper, we study the question of extendability of these two properties to split permutation graphs. We answer this question negatively with respect to both properties. Moreover, we conjecture that with respect to both of them the split permutation graphs constitute a critical class.