On pairwise compatibility graphs having Dilworth number two

It is known that LPG∩mLPG is not empty and that threshold graphs, i.e. Dilworth one graphs, are contained in it. In this paper we prove that Dilworth two graphs belong to the set LPG∩mLPG, too. Our proof is constructive since we show how to compute all the parameters T, w, dmax and dmin exploiting the usual representation of Dilworth two graphs in terms of node weight function and thresholds. For graphs with Dilworth number two that are also split graphs, i.e. split permutation graphs, we provide another way to compute T, w, dmin and dmax when these graphs are given in terms of their intersection model.