THE NICHE GRAPHS OF INTERVAL ORDERS

The niche graph of a digraph D is the (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if N-D(+)(x) boolean AND N-D(+)(y) not equal theta or N-D(-) (x) boolean AND N-D(-)(y) not equal theta, where N-D(+)(x) (resp. N-D(+)(x)) is the set of out-neighbors (resp. in-neighbors) of x in D. A digraph D = (V, A) is called a semiorder (or a unit interval order) if there exist a real-valued function f : V -> R on the set V and a positive real number delta is an element of R such that (x, y) E A if and only if f (x) > f (y) + delta digraph D = (V, A) is called an interval order if there exists an assignment J of a closed real interval J(x) c N to each vertex x E V such that (x, y) is an element of A if and only if min J(x) > max J(y).