# The Toucher-Isolator game

## TL;DR

This paper introduces the Toucher-Isolator game, a new positional game on graphs where players aim to maximize or minimize touched vertices, analyzing optimal strategies and outcomes for various graph classes.

## Contribution

It presents the first analysis of the Toucher-Isolator game, including optimal play and bounds for different graph types, expanding the study of Maker-Breaker style games.

## Key findings

- Determined the number of untouched vertices for general graphs.
- Provided tight bounds and examples for cycles, paths, and trees.
- Analyzed game outcomes on k-regular graphs.

## Abstract

We introduce a new positional game called `Toucher-Isolator', which is a quantitative version of a Maker-Breaker type game. The playing board is the set of edges of a given graph G, and the two players, Toucher and Isolator, claim edges alternately. The aim of Toucher is to `touch' as many vertices as possible (i.e. to maximise the number of vertices that are incident to at least one of her chosen edges), and the aim of Isolator is to minimise the number of vertices that are so touched. We analyse the number of untouched vertices u(G) at the end of the game when both Toucher and Isolator play optimally, obtaining results both for general graphs and for particularly interesting classes of graphs, such as cycles, paths, trees, and k-regular graphs. We also provide tight examples.