Characterization of Split Comparability Graphs

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A split comparability graph is a split graph which is transitively orientable. In this work, we characterize split comparability graphs in terms of vertex labelling. Further, using this characterization, we prove that the permutation-representation number of a split comparability graph is at most three. This gives us an alternative proof of the result in order theory that the dimension of a split order is at most three.