What is the cycle structure of a graph?

I have a vague imagination of what the cycle structure of a graph might be - something taking into account the numbers, lengths, Hamiltonianicities, Eulerianicities and whatsoever of cycles of a graph and their quantified interweavings - and there are of course papers and books mentioning the cycle structure of graphs - see Quo vadis, graph theory?, for example - but I cannot find a tangible definition to start with.

Question: What might be a sensible definition of "cycle structure"?

-
This is very much not a real question as it stands... You surely agree? –  Mariano Suárez-Alvarez Sep 21 '10 at 23:21
On the one hand I agree; on the other, I have an answer (written below). –  David Speyer Sep 21 '10 at 23:24
If you ask me: of course it is a real question! Should I re-formulate it? It is a matter of fact that the term "cycle structure" is in use, and I am looking for a definition. Maybe I am mislead and it does not make sense to look for such a definition? –  Hans Stricker Sep 21 '10 at 23:27
Hans, the text you wrote does not ask anything. What do you want to know? –  Mariano Suárez-Alvarez Sep 21 '10 at 23:28
I edited the text, now there is a question. Sorry for that. –  Hans Stricker Sep 21 '10 at 23:37
show 1 more comment

This is a vague question, but here is an attempt at an answer. Let $G$ be a graph, let $E$ be the set of edges of $G$, and let $C \subset 2^E$ be the set of cycles of $G$. Then knowing $C$ is equivalent to the matroid of $G$. Two graphs produce the same matroid if and only if they are related by a sequence of the following moves:

(1) Taking two connected components and gluing them along a single vertex, or undoing the above.

(2) If $G$ has two vertices $u$ and $v$ so that $G \setminus \{ u,v \}$ is disconnected, cutting along those vertices and regluing some of the pieces back with $u$ and $v$ switched.

In particular, if a graph is $3$-connected, then it is determined by its matroid.

So one answer could be "The cycle structure of a graph is its matroid" and, as the above shows, this contains almost as much information as the graph.

-
I'll just comment that I agree that this is what cycle structure of a graph means. Also, the theorem that David mentions was first proven by Whitney. –  Tony Huynh Sep 21 '10 at 23:29
This sounds very promising. But if this was a real answer, why isn't my question a real question? –  Hans Stricker Sep 21 '10 at 23:36