The Petersen family. K6 is at the top of the illustration, K3,3,1 is in the upper right, and the Petersen graph is at the bottom. The blue links indicate ΔY- or YΔ-transforms between graphs in the family.
In graph theory, the Petersen family is a set of seven undirected graphs that includes the Petersen graph and the complete graph K6. The Petersen family is named after Danish mathematician Julius Petersen, the namesake of the Petersen graph.
Any of the graphs in the Petersen family can be transformed into any other graph in the family by YΔ- and ΔY-transformations, operations in which a triangle is replaced by a degree-three vertex or vice versa. These seven graphs form the forbidden minors for linklessly embeddable graphs, graphs that can be embedded into three-dimensional space in such a way that no two cycles in the graph are linked.[1] They are also among the forbidden minors for the YΔY-reducible graphs.[2][3]
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