Pebble exchange group of graphs

A graph puzzle Puz(G) of a G is defined as follows. configuration bijection from the set vertices board to pebble graph, both graphs being isomorphic some input G. move pebbles exchanging two which are adjacent on and graph. For pair configurations f g, we say that equivalent g if can be transformed into by finite sequence moves. Let Aut(G) automorphism group G, let 1G unit element Aut(G). The exchange denoted Peb(G), all automorphisms such each other. In this paper, basic properties Peb(G) studied. Among other results, it shown for any connected contained in Peb(G2), where G2 square