Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The case for $\operatorname{cr}(G)=0$ (planar graphs) is given by Wagner's theorem, but what about (for instance) the family of graphs with $\operatorname{cr}(G) \leq 1$?

share|cite|improve this question
up vote 2 down vote accepted

Check out this link. It claims that the family of graphs with crossing number $\leq k$ is not minor-closed for general $k$, though I don't understand the example there. One can fix the definition so that it be minor-closed by "blatantly" enforcing it, see here. Of course, once we have a minor-closed family, Robertson-Seymour theory tells us that there is a finite list of forbidden minors.

share|cite|improve this answer
Very helpful, thank you. I understand the argument given in the first link for the claim that $cr(G/e) \nleq cr(G)$ in general, which means that the families I described are not minor-closed. – thefringthing Sep 10 '11 at 21:58

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.