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)∩N + D (y) 6= ∅ or N − D (x)∩ N D (y) 6= ∅, 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 δ ∈ R such that (x, y) ∈ A if and only if f(x) > f(y) + δ. A digraph D = (V,A) is called an interval order if there exists an assignment J of a closed real interval J(x) ⊂ R to each vertex x ∈ V such that (x, y) ∈ A if and only if min J(x) > maxJ(y). S. -R. Kim and F. S. Roberts characterized the competition graphs of semiorders and interval orders in 2002, and Y. Sano characterized the competition-common enemy graphs of semiorders and interval orders in 2010. In this note, we give characterizations of the niche graphs of semiorders and interval orders.