On the partial order competition dimensions of chordal graphs

TL;DR

This paper investigates the partial order competition dimensions of chordal graphs, establishing upper bounds for diamond-free cases and demonstrating the existence of chordal graphs with higher dimensions when diamonds are present.

Contribution

It proves that diamond-free chordal graphs have partial order competition dimensions at most three and shows that some chordal graphs with diamonds exceed this bound.

Findings

Diamond-free chordal graphs have dimension ≤ 3

Existence of chordal graphs with diamonds and dimension > 3

Partial order competition dimension relates to graph structure

Abstract

Choi {\it et al.} [{J.~Choi, K.~S.~Kim, S.~-R.~Kim, J.~Y.~Lee, and Y.~Sano}: {On the competition graphs of $d$-partial orders}, \emph{Discrete Applied Mathematics} (2015), \texttt{http://dx.doi.org/10.1016/j.dam.2015.11.004}] introduced the notion of the partial order competition dimension of a graph. It was shown that complete graphs, interval graphs, and trees, which are chordal graphs, have partial order competition dimensions at most three. In this paper, we study the partial order competition dimensions of chordal graphs. We show that chordal graphs have partial order competition dimensions at most three if the graphs are diamond-free. Moreover, we also show the existence of chordal graphs containing diamonds whose partial order competition dimensions are greater than three.