Pebble Exchange Group of Graphs

Abstract

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.