The niche graphs of interval orders

TL;DR

This paper characterizes the structure of niche graphs derived from semiorders and interval orders, expanding understanding of their properties in graph theory and their relation to specific classes of digraphs.

Contribution

It provides new characterizations of niche graphs for semiorders and interval orders, building on prior work on competition and competition-common enemy graphs.

Findings

Characterizations of niche graphs for semiorders

Characterizations of niche graphs for interval orders

Extension of previous graph class characterizations

Abstract

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 $\delta \in \mathbb{R}$ such that $(x,y) \in A$ if and only if $f(x) > f(y) + \delta$. 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.