The most important feature of intervals of the real line is their ordering. This relation is obviously irreflexive, asymmetric, and transitive, i.e. a partial order. The reason is that not every poset can be represented by intervals on the real line. Those posets that can are called interval orders .
Interval graphs are exactly the cocomparability graphs of interval orders.
There are various other geometric intersection models where such an order is apparent. In all such classes we may use the important fact that cocomparability graphs are perfect, and many parameters can be computed efficiently for them. A graph is a permutation graph if and only if both G and its complement are comparability graphs. But the 5-cycle of the example is no cocomparability graph, it is even not perfect. Thus the pattern is impossible.