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 $N^+_D(x) \cap N^+_D(y) \neq \emptyset$ or $N^-_D(x) \cap N^-_D(y) \neq \emptyset$, 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 $f:V \to \mathbb{R}$ on the set $V$ and a positive real number $δ\in \mathbb{R}$ such that $(x,y) \in A$ 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 $J(x) \subset \mathbb{R}$ to each vertex $x \in V$ such that $(x,y) \in A$ if and only if $\min J(x) > \max J(y)$. 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.

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