The relationship between the threshold dimension of split graphs and various dimensional parameters

Let the coboxicity of a graph G be denoted by cob(G), and the threshold dimension by t(G). For fixed k?3, determining if cob(G)?k and t(G)?k are both NP-complete problems. We show that if G is a comparability graph, then we can determine if cob(G)?2 in polynomial time. This result shows that it is possible to determine if the interval dimension of a poset equals 2 in polynomial time. If the clique covering number of G is 2, we show that one can determine if t(G)?2 in polynomial time. Sufficient conditions on G are given for cob(G)?2 and for t(G)?2.