The Petersen family of graphs, not to be confused with generalized Petersen graphs, are a set of seven graphs obtained from the Petersen
graph (or complete graph 
) by 
-
or 
-
transforms.
Here, the 
-
transform corresponds to replacing the
three graph edges forming a triangle
graph 
are by three graph edges and a new graph
vertex that form a 
, and the 
-
transform to the reverse operation of this.
As illustrated above and enumerated in the following table, the Petersen family graphs include the Petersen graph 
, complete tripartite
graph 
,
complete graph 
, and complete bipartite
graph minus edge 
.
| index | vertex count | graph |
| 1 | 10 | Petersen
graph  |
| 2 | 9 | Petersen
family graph  |
| 3 | 8 | Petersen
family graph  |
| 4 | 7 | Petersen
family graph  |
| 5 | 7 | complete
tripartite graph  |
| 6 | 6 | complete graph  |
| 7 | 8 | complete
bipartite graph minus edge  |
Sachs (1983) showed that the seven graphs of the Petersen family are intrinsically linked, i.e., no matter how they are embedded in space, they have cycles that are linked to each other. He also suggested characterization of these graphs via forbidden subgraphs. Robertson et al. (1993) resolved this question by establishing that intrinsically linked graphs are characterized by having a member of the Petersen family as a graph minor.
In addition, the Petersen family graphs are among the forbidden minors of apex graphs.
