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