  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 $G$.
