Tolerance orders and bipartite unit tolerance graphs
We construct all six-element orders which are not 50%-tolerance orders. We show that a width-two order is a 50% tolerance order if and only if no restriction of the order to a six-element set is isomorphic to one of these six-element orders. This yields a corresponding characterization of bipartite 50%-tolerance graphs. Since an order (graph) has a 50% tolerance representation if and only if it has a unit tolerance representation, our results apply to unit tolerance orders (graphs) as well.
Bogart, K. P.,
Jacobson, M. S.,
McMorris, F. R.,
Langley, L. L.
Tolerance orders and bipartite unit tolerance graphs.
Discrete Mathematics, 226(1--3), 35–50.