The fixing number 
of a finite graph 
is the smallest cardinality of a fixing set of 
, where a fixing set is a vertex set 
whose stabilizer is trivial (Erwin and Harary
2006, Gibbons and Laison 2009).
For a vertex 
of 
, the stabilizer of 
, 
,
is the set of group elements 
, where 
is the graph automorphism
group. The vertex stabilizer of a set of vertices 
is then defined as 
. A vertex 
is fixed by a group element 
if 
.
Fixing numbers are integers that range from 0 (for an identity graph) to 
(for a complete or empty
graph), where 
is a graph's vertex count.
The fixing number of a graph is equal to that of its graph complement.
Greenfield (2011) summarizes values of the fixing numbers for a number of graph families and discusses a number of fixing number algorithms.
Gibbons and Laison (2009) proposed a greedy algorithm for determining a fixing number, noting that it was not known to be well-defined. The 12-vertex Greenfield graph was subsequently discovered by Greenfield (2011), demonstrating that the result of the algorithm can depend on the vertices chosen at each step and so establishing that it provides only an upper limit to the graph's fixing number. Greenfield's graph (which Greenfield postulated to be the smallest exceptional graph possible), together with a number of other such exceptional graphs found by E. Weisstein on Jun. 6, 2023 and Aug. 10-11 and 21, 2026, are summarized in the table below. The first few such graphs are illustrated above.
| vertex count | fixing number | greedy fixing number | graph |
| 12 | 3 | 4 | Greenfield graph |
| 16 | 2 | 3 | 16-cubic graph 3972 |
| 16 | 2 | 3 |  -regular nonplanar diameter graph |
| 20 | 5 | 7 | Folkman graph |
| 84 | 4 | 5 | Kneser
graph  |
| 84 | 4 | 5 | tetrahedral Johnson graph  |
| 121 | 8 | 9 |  antelope graph |
| 168 | 4 | 5 | bipartite Kneser graph  |
| 210 | 4 | 5 | Johnson
graph  |
| 210 | 4 | 5 | Kneser graph  |
| 330 | 5 | 6 | bipartite
Kneser graph  |
| 420 | 4 | 5 | bipartite Kneser graph  |
However, excepting a small number of such cases, the greedy algorithm does give the actual fixing number for almost all small named or tabulated simple graphs.
