User:MWinter4/Heawood family

In graph theory the name Heawood family is used to refer to two related graph families that are closed under ΔY- and YΔ-transformations:
 * the family of 20 graphs generated from the complete graph $$K_7$$.
 * the family of 78 graphs generated from $$K_7$$ and $$K_{3,3,1,1}$$.

The members of either Heawood family are sometimes called Heawood graphs, though this has the potential for confusion with the Heawood graph. The Heawood graph is a however a members of both families. This is in analogy to the Petersen family, which too is named after a notable member – the Petersen graph.

The Heawood families play a major role in topological graph theory. No member of either family is 4-flat, and no members of the $$K_7$$-family is knotless.

No member of the Heawood family is 4-flat, and therefore neither linkless nor planar. All members have Colin de Verdière invariant $$\mu=6$$. It is conjectured that the family of Heawood graphs is the complete list of excluded minors for both the 4-flat graphs and the graphs with $$\mu\le 5$$.

It is known that neither the Heawood family nor the $$K_7$$-family gives the complete list of excluded minors for the knotless graphs.

The $$K_7$$-family
The $$K_7$$-family plays an important role in the study of knotless graphs. All members of the family are excluded minors for the class of knotless graphs, but they are not a complete list.

The $$\{K_7,K_{3,3,1,1}\}$$-family
The Heawood family plays an important role in the study of 4-flat graphs. The Heawood graphs are the only excluded minors for the 4-flat graphs.

Background
The Heawood family naturally generalize the excluded minors for planar and linkless graphs. For all of those the list of excluded minors is closed under ΔY- and YΔ-transformations. The generators are:
 * for planar, $$K_5$$ and $$K_{3,3}$$ (the Kuratowski graphs).
 * for linkless, $$K_6$$ and $$K_{3,3,1}$$ (generate the Petersen family).
 * (conjecturally) for 4-flat, $$K_7$$ and $$K_{3,3,1,1}$$ (generate the Heawood family).

For planar and linkless graphs it is actually sufficient to generate the excluded minors from only $$K_5$$ and $$K_6$$ respectively. For the Heawood family however both generators are necessary. The family generated by $$K_7$$ has 20 members and the family generated by $$K_{3,3,1,1}$$ has 58 members. The union of these disjoint families yields the Heawood family.