# Pebble Exchange Group of Graphs

## TL;DR

This paper introduces the pebble exchange group of a graph, studying its properties and showing that for any connected graph, all automorphisms are contained in the pebble exchange group of its square graph.

## Contribution

The paper defines the pebble exchange group of a graph and explores its properties, including the result that all automorphisms are in the group for the square of any connected graph.

## Key findings

- All automorphisms of a connected graph are in the pebble exchange group of its square graph.
- Basic properties of the pebble exchange group are established.
- The relationship between automorphisms and pebble exchanges is characterized.

## Abstract

A graph puzzle ${\rm Puz}(G)$ of a graph $G$ is defined as follows. A configuration of ${\rm Puz}(G)$ is a bijection from the set of vertices of a board graph to the set of vertices of a pebble graph, both graphs being isomorphic to some input graph $G$. A move of pebbles is defined as exchanging two pebbles which are adjacent on both a board graph and a pebble graph. For a pair of configurations $f$ and $g$, we say that $f$ is equivalent to $g$ if $f$ can be transformed into $g$ by a finite sequence of moves. Let ${\rm Aut}(G)$ be the automorphism group of $G$, and let ${\rm 1}_G$ be the unit element of ${\rm Aut}(G)$. The pebble exchange group of $G$, denoted by ${\rm Peb}(G)$, is defined as the set of all automorphisms $f$ of $G$ such that ${\rm 1}_G$ and $f$ are equivalent to each other. In this paper, some basic properties of ${\rm Peb}(G)$ are studied. Among other results, it is shown that for any connected graph $G$, all automorphisms of $G$ are contained in ${\rm Peb}(G^2)$, where $G^2$ is a square graph of $G$.