Is the 3-sum of two graphic matroids a graphic matroid?

Question

A regular matroid is a matroid which is representable over any field. It is a famous theorem of Seymour's that the any regular matroid is obtained by performing 1,2, and 3 sums on graphic, cographic or R10 matroids.

Clearly then, the operations of 1,2 and 3 sum preserve regularity.

My question is whether a stronger statement holds: Is it true that the $k$-sum of two graphic matroids is again a graphic matroid, for $k \in \{1,2,3\}$? I have the same question about whether cographicness is preserved.

Finally, what about the larger class of "network matroids" (a matroid which is either graphic or cographic). Is this class closed under the $k$-sum operation for $1 \leq k \leq 3$?

These are definitely known results, but I am having a hard time pinning down the answers in the literature. I am hoping someone more familiar with the area can lend a hand. Thanks!

Answer 1

Graphicness is closed under all of these kinds of generalised parallel connections. This is because you can take representative graphs for the summands and perform this operation on the graphs. The cycle matroid of the resulting graph will be the (k-)sum of the original matroids. For cographic matroids, I think you can perform a dual construction. A dual version of the 3-sum would involve identifying bonds of size 3 and then contracting the identified edges. I'm not sure about network matroids in general.