A notion introduced by R. M. Wilson in 1974. Given a finite graph G with n vertices, puz(G) is defined as the graph whose nodes are the labelings of G leaving one node unoccupied, i.e., the ways to place n-1 different counters on n-1 nodes of G. This labelings can be identified with the permutations of {0,1,2,...,n-1}, so that puz(G) has n! nodes. Two labelings are connected by an edge in puz(G) iff one can be transformed into the other by moving one of the labels along one edge of G.


The possible labelings of two vertices of the path graph P_3 are illustrated above, giving puz(P_3) as illustrated.

If G is the square graph C_4, then puz(C_4) consists of two disjoint cycles with 12 nodes. In general, the puz-graph of an n-cycle graph has (n-2)! connected components, each having n(n-1) nodes (Vajda 1992). Wilson proved that the puz-graph of a finite simple biconnected graph G that is not polygonal always has two connected components if G is bipartite. Otherwise, with one surprising exception, puz(G) is connected. The exception is the puz-graph of the theta-0 graph, which surprisingly has six connected components.

The paths connecting two labelings L_1 and L_2 in puz(G) represent the sequences of moves that take L_1 to L_2. Hence, these can be transformed into each other if and only if they belong to the same connected component of puz(G). In most of the cases, this cannot be decided by looking at puz(G), which almost always has too many nodes to be adequate for practical use. This problem is solved using a criterion by Wilson, which can be easily expressed in terms of G, L_1 and L_2: L_1 and L_2 are linked by a sequence of moves if and only if the distance between their unoccupied nodes and the permutation taking L_1 to L_2 are either both even or both odd.


Wilson's criterion can be applied to the 15 puzzle as follows. Each arrangement of the 15 squares corresponds to a labeling of 15 nodes of the grid graph G_(4,4). Since G_(4,4) is bipartite, puz(G_(4,4)) is disconnected, so the puzzle does not always have a solution. This can be seen by looking at the labelings of the 15 puzzle configurations illustrated above. The distance between the unoccupied nodes is 0, but the permutation taking one labeling to the other is the cycle (1 2), which is odd. Hence it is impossible to solve the puzzle starting from the configuration at right.

The Hanoi graph H_n is the puz-graph of the possible configurations of n towers of Hanoi. Since it is connected, the game always has a solution.

See also

15 Puzzle, Graph, Hanoi Graph, Permutation, Tower of Hanoi

This entry contributed by Margherita Barile

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Vajda, S. Mathematical Games and How to Play Them. Chichester, England: Ellis Horwood, pp. 1-2, 1992.Wilson, R. M. "Graph Puzzles, Homotopy, and the Alternating Group." J. Combin. Th. B 16, 86-96, 1974.

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Cite this as:

Barile, Margherita. "Puz-Graph." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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