On the competition graphs of $d$-partial orders

TL;DR

This paper characterizes competition graphs of $d$-partial orders, shows any graph can be realized as such with isolated vertices, and explores graphs with low partial order competition dimensions.

Contribution

It extends previous characterizations of competition graphs, introduces the partial order competition dimension, and analyzes graphs with dimensions up to three.

Findings

Characterization of competition graphs of $d$-partial orders.

Any graph can be a competition graph of a $d$-partial order with isolated vertices.

Study of graphs with partial order competition dimension at most three.

Abstract

In this paper, we study the competition graphs of $d$-partial orders and obtain their characterization which extends results given by Cho and Kim \cite{chokim} in 2005. We also show that any graph can be made into the competition graph of a $d$-partial order for some positive integer $d$ as long as adding isolated vertices is allowed. We then introduce the notion of the partial order competition dimension of a graph and study graphs whose partial order competition dimensions are at most three.