Split comparability graphs are exactly the split graphs whose clique can be labeled so that each independent-set vertex sees a prefix, a suffix, or a two-ended interval, and every such graph has permutation-representation number at most three.
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Characterization of Split Comparability Graphs
Split comparability graphs are exactly the split graphs whose clique can be labeled so that each independent-set vertex sees a prefix, a suffix, or a two-ended interval, and every such graph has permutation-representation number at most three.