# On the isolation number of graphs with minimum degree four

## TL;DR

This paper introduces a new technique for establishing upper bounds on the isolation number in graphs with minimum degree four, including special bounds for triangle-free graphs, advancing understanding of vertex-edge domination.

## Contribution

The paper presents a novel method for bounding the isolation number in graphs with minimum degree four, including specific bounds for triangle-free graphs.

## Key findings

- For graphs with minimum degree at least 4, the isolation number is at most 13n/41.
- For triangle-free graphs with minimum degree at least 4, the bound improves to 3n/10.
- The technique provides a systematic way to derive upper bounds on the isolation number.

## Abstract

An isolating set in a graph $G$ is a set $S$ of vertices such that removing $S$ and its neighborhood leaves no edge. The isolation number $\iota(G)$ of $G$ (also known as the vertex-edge domination number) is the minimum size among all isolating sets of $G$. We provide a technique for proving upper bounds on this parameter for graphs with a given minimum degree. For example, we show that if $G$ has order~$n$ and minimum degree at least~$4$, then $\iota(G) \le 13n/41$, and if $G$ is also triangle-free, then $\iota(G) \le 3n/10$.