Possible error on Wiki-page for Dual_matroid

Question

On the Wiki-page https://en.wikipedia.org/wiki/Dual_matroid you find the following part:

It reads "..the graphic matroids of planar graphs are self-dual". This claim cannot be right. Do you agree? Or am I mistaken?

Answer 1

The sentence in question is correct, if rather misleading when not taken in context. The key here is the definition of the "self-dual family of matroids" given in the first sentence:

Self-dual matroid family: a family $\mathscr{F}$ of matroids such that $M \in \mathscr{F}$ $\Longleftrightarrow$ $M^*$ is also in $\mathscr{F}$.

The sentence in question is misleading because it is making a statement about a family of matroids without explicitly using the word family. The Wikipedia entry has now been edited to read

Thus, the family of graphic matroids of planar graphs is self-dual.

Now, let's see why that is true. Let $\mathscr{P}$ be the family of all matroids arising from planar graphs and let $M \in \mathscr{P}$. Then there is planar graph $G$ such that $M=M(G)$. Since $G$ is planar, so is its dual graph $G^*$. Now consider the matroid $M(G^*)$. From, e.g., the circuit axioms of a matroid one can deduce that $M(G^*)$ is isomorphic to $M^*$, the dual matroid of $M$.