Induced Saturation of $P_{6}$

TL;DR

This paper proves the existence of a $P_{6}$-induced-saturated graph, filling a gap in the understanding of induced saturation properties for paths of length six.

Contribution

It demonstrates for the first time that a $P_{6}$-induced-saturated graph exists, extending previous results known for shorter paths.

Findings

Existence of a $P_{6}$-induced-saturated graph

Fills a gap in induced saturation theory for paths

Builds on prior work showing non-existence for shorter paths

Abstract

A graph $G$ is called $H$-induced-saturated if $G$ does not contain an induced copy of $H$, but removing any edge from $G$ creates an induced copy of $H$ and adding any edge of $G^{c}$ to $G$ creates an induced copy of $H$. Martin and Smith showed that there does not exist a $P_{4}$-induced-saturated graph, where $P_{4}$ is the path on 4 vertices. Axenovich and Csik\'os studied related questions, and asked if there exists a $P_{n}$-induced-saturated graph for any $n\geq5$. Our aim in this short note is to show that there exists a $P_{6}$-induced-saturated graph.