Petersen family

Although these operations can in principle lead to multigraphs, that does not happen within the Petersen family.

[1] Horst Sachs had previously studied such embeddings, shown that the seven graphs of the Petersen family do not have such embeddings, and posed the question of characterizing the linklessly embeddable graphs by forbidden subgraphs.

A connected graph is YΔY-reducible if it can be reduced to a single vertex by a sequence of steps, each of which is a ΔY- or YΔ-transform, the removal of a self-loop or multiple adjacency, the removal of a vertex with one neighbor, and the replacement of a vertex of degree two and its two neighboring edges by a single edge.

For instance, the complete graph K4 can be reduced to a single vertex by a YΔ-transform that turns it into a triangle with doubled edges, removal of the three doubled edges, a ΔY-transform that turns it into the claw K1,3, and removal of the three degree-one vertices of the claw.

[2] In fact, as Yaming Yu showed, there are at least 68,897,913,652 forbidden minors for the YΔY-reducible graphs beyond the seven of the Petersen family.