# Isolation of $k$-cliques

## TL;DR

This paper introduces a bound on the minimum size of a vertex set whose removal eliminates all $k$-cliques from a connected graph, generalizing classical and recent results and solving an open problem.

## Contribution

It establishes a sharp upper bound on the size of such vertex sets for any positive integer $k$, extending previous specific cases.

## Key findings

- Bound is tight for general graphs
- Includes classical and recent special cases
- Solves an open problem in graph theory

## Abstract

For any positive integer $k$ and any $n$-vertex graph $G$, let $\iota(G,k)$ denote the size of a smallest set $D$ of vertices of $G$ such that the graph obtained from $G$ by deleting the closed neighbourhood of $D$ contains no $k$-clique. Thus, $\iota(G,1)$ is the domination number of $G$. We prove that if $G$ is connected, then $\iota(G,k) \leq \frac{n}{k+1}$ unless $G$ is a $k$-clique or $k = 2$ and $G$ is a $5$-cycle. The bound is sharp. The case $k=1$ is a classical result of Ore, and the case $k=2$ is a recent result of Caro and Hansberg. Our result solves a problem of Caro and Hansberg.