Coordinating Pebble Motion On Graphs, The Diameter Of Permutation Groups, And Applications
SFCS '84 Proceedings of the 25th Annual Symposium onFoundations of Computer Science, 1984
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Reconfigurations in graphs and grids
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
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Let r, n and n"1,...,n"r be positive integers with n=n"1+...+n"r. Let X be a connected graph with n vertices. For 1@?i@?r, let P"i be the i-th color class of n"i distinct pebbles. A configuration of the set of pebbles P=P"1@?...@?P"r on X is defined as a bijection from the set of vertices of X to P. A move of pebbles is defined as exchanging two pebbles with distinct colors on the two endvertices of a common edge. For a pair of configurations f and g, we write f~g if f can be transformed into g by a sequence of finite moves. The relation ~ is an equivalence relation on the set of all the configurations of P on X. We study the number c(X,n"1,...,n"r) of the equivalence classes. A tuple (X,n"1,...,n"r) is called transitive if for any configuration f and for any vertex u, a pebble f(u) can be moved to any other vertex by a sequence of finite moves. We determine c(X,n"1,...,n"r) for an arbitrary transitive tuple (X,n"1,...,n"r).