Pebble Exchange Group of Graphs
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A graph puzzle Puz(G) of a graph G is defined as follows. A configuration of Puz(G) is a bijection from the set of vertices of a board graph G to the set of vertices of a pebble 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 sequence of finite moves. Let Aut(G) be the automorphism group of G, and let 1_G be the unit element of Aut(G). The pebble exchange group of G, denoted by Peb(G), is defined as the set of all automorphisms f of G such that 1_G and f are equivalent to each other. In this paper, some basic properties of Peb(G) are studied. Among other results, it is shown that for any connected graph G, all automorphisms of G are contained in Peb(G^2), where G^2 is a square graph of G.
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