Q:

Can any undirected connected graph (UCG) with $N$ cycles be decomposed as 2 UCG with $N-1$ cycles?

Consider any (arbitrary) undirected connected graph $\mathcal{G}_{AB} = (V,E_{AB})$ which has $N$ cycles, $V$ is the set of vertices and $E_{AB}$ the set of edges.
I'm wondering if it is always possible to decompose $\mathcal{G}_{AB}$ as the superposition (union?) of two undirected connected graphs $\mathcal{G}_{A}=(V,E_{A})$ and $\mathcal{G}_B=(V,E_{B})$ which both have $N-1$ cycles?
Maybe its redundant to say it (since all graphs are connected) but note that $\mathcal{G}_{AB},\mathcal{G}_A,\mathcal{G}_B$ all share the same vertex set $V$.
If the answer is no: is it possible to decompose $\mathcal{G}_{AB}$ with $\mathcal{G}_A, \mathcal{G}_B$ with $N_A,N_B<N$ cycles respectively?
Thanks in advance!
EDIT:
By decomposition, I don't mean that $E_A$ and $E_B$ are disjoint, as in the usual sense. I mean that $E_A,E_B$ may have non empty intersection and that $E_{AB}=E_A\cup E_B$.

A:

Assuming that by the number of cycles you mean the dimension of the cycle space: for a connected graph, that dimension is just $m-n+1$ (source: Wikipedia), where $m$ is the number of edges and $n$ is the number of vertices.
If $G$ is a connected graph with $m-n+1>0$, then it has at least one cycle. Pick two edges $e, e'$ on that cycle; let $G_A = G - e$ and $G_B = G - e'$. Then $G = G_A \cup G_B$ and each of them has a cycle space of dimension one less than in $G$.

