# On induced saturation for paths

## TL;DR

This paper investigates the existence of graphs that are saturated with respect to induced paths, proving the existence of infinitely many such graphs for paths of lengths multiple of three and analyzing specific cases like Kneser graphs.

## Contribution

It establishes the existence of infinitely many $P_{3n}$-induced-saturated graphs and explores the properties of Kneser graphs in this context.

## Key findings

- Existence of $P_{3n}$-induced-saturated graphs for all positive integers n.
- Construction of infinitely many such graphs for each $P_{3n}$.
- Kneser graphs $K(n,2)$ are $P_6$-induced-saturated for all $n  5.

## Abstract

For a graph $H$, a graph $G$ is $H$-induced-saturated if $G$ does not contain an induced copy of $H$, but either removing an edge from $G$ or adding a non-edge to $G$ creates an induced copy of $H$. Depending on the graph $H$, an $H$-induced-saturated graph does not necessarily exist. In fact, Martin and Smith (2012) showed that $P_4$-induced-saturated graphs do not exist, where $P_k$ denotes a path on $k$ vertices. Axenovich and Csik\'{o}s (2019) asked the existence of $P_k$-induced-saturated graphs for $k \ge 5$; it is easy to construct such graphs when $k\in\{2, 3\}$. Recently, R\"{a}ty constructed a graph that is $P_6$-induced-saturated. In this paper, we show that there exists a $P_{k}$-induced-saturated graph for infinitely many values of $k$. To be precise, we find a $P_{3n}$-induced-saturated graph for every positive integer $n$. As a consequence, for each positive integer $n$, we construct infinitely many $P_{3n}$-induced-saturated graphs. We also show that the Kneser graph $K(n,2)$ is $P_6$-induced-saturated for every $n\ge 5$.