Interval order

Interval order --- интервальный порядок

1. An ordering [math]\displaystyle{ (X_{1}, \ldots, X_{n}) }[/math] of the maximal cliques of a graph [math]\displaystyle{ G }[/math] such that for every vertex the maximal cliques containing it occur consecutively in the ordering is called interval order.

2. Let [math]\displaystyle{ I = (I_{j})_{j=1}^{n} }[/math] be a finite collection of intervals of the real line and let [math]\displaystyle{ (I,L) }[/math] be a poset such that [math]\displaystyle{ I_{i}LI_{j} }[/math] iff [math]\displaystyle{ I_{i} }[/math] is completely to the left of [math]\displaystyle{ I_{j} }[/math]. [math]\displaystyle{ (P,\lt ) }[/math] is an interval order if there is a collection [math]\displaystyle{ I }[/math] of intervals such that [math]\displaystyle{ (P,\lt ) }[/math] is isomorphic to [math]\displaystyle{ (I,L) }[/math]. It is easy to see that the comparability graphs of interval orders are exactly the cointerval graphs.