Competition graphs induced by permutations

TL;DR

This paper explores competition graphs derived from permutations, showing their equivalence to those from doubly partial orders, and classifies graphs from specific permutation patterns, with initial results on weighted variants.

Contribution

It demonstrates the equivalence of competition graphs from permutations and doubly partial orders, classifies graphs from 132-avoiding permutations, and introduces initial findings on weighted competition graphs.

Findings

Competition graphs from permutations are equivalent to those from doubly partial orders.

Graphs from 132-avoiding permutations avoid induced path graphs of length 3.

Initial enumerative and structural results are provided for weighted competition graphs.

Abstract

In prior work, Cho and Kim studied competition graphs arising from doubly partial orders. In this article, we consider a related problem where competition graphs are instead induced by permutations. We first show that this approach produces the same class of competition graphs as the doubly partial order. In addition, we observe that the $123$ and $132$ patterns in a permutation induce the edges in the associated competition graph. We classify the competition graphs arising from $132$-avoiding permutations and show that those graphs must avoid an induced path graph of length 3. Finally, we consider the weighted competition graph of permutations and give some initial enumerative and structural results in that setting.