Inducibility of 4-vertex tournaments

TL;DR

This paper determines the inducibility of all small tournaments with up to four vertices and describes their extremal constructions, highlighting a unique case involving an unbalanced blow-up and quasi-random components.

Contribution

It provides a complete characterization of the inducibility for all 4-vertex tournaments and introduces novel extremal constructions for these cases.

Findings

Complete inducibility values for all 4-vertex tournaments

Identification of a unique extremal construction involving quasi-random tournaments

Description of an iterative blow-up construction for extremal cases

Abstract

We determine the inducibility of all tournaments with at most $4$ vertices together with the extremal constructions. The $4$-vertex tournament containing an oriented $C_3$ and one source vertex has a particularly interesting extremal construction. It is an unbalanced blow-up of an edge, where the sink vertex is replaced by a quasi-random tournament and the source vertex is iteratively replaced by a copy of the construction itself.