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

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

The paths connecting two labelings and in
represent the sequences of moves that take to .
Hence, these can be transformed into each other if and only if they belong to the
same connected component of .
In most of the cases, this cannot be decided by looking at , 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 ,
and : and are linked by a sequence of moves if and only if the distance
between their unoccupied nodes and the permutation
taking
to 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 . Since is bipartite, 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 is the puz-graph of the possible configurations of towers of
Hanoi. Since it is connected, the game always has a solution.

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. B16, 86-96, 1974.