Circuits and Trees Given a graph G, can it be split into 2 sets of graphs($ G_1, \; G_2 $) such that, $G_1$ consists only trees and $G_2$ consists only circuits ?
In other words: Is it possible to construct any graph using only circuits and trees ? And if so, what are some examples to the contrary ?
 A: An important distinction to draw is that circuits are closed trails (repeated vertices allowed within a trail) and that cycles are closed simple paths (no repeated vertices allowed).
Then it is possible to decompose (an undirected finite graph) $G$ into edge disjoint subgraphs $G_1$ which is a vertex disjoint collection of trees (a forest) and $G_2$ which is a vertex disjoint collection of circuits.
One way to proceed is by a method of descent (induction).  If $G$ does not contain a cycle, then it is a forest and we are done ($G_1 = G$ and $G_2 = \emptyset$).
As long as $G$ does contain a cycle, say $C$, remove those edges and any vertices otherwise isolated in $G$ to get $G\backslash C$.  By induction hypothesis $G\backslash C$ can be decomposed into a forest $G_1$ and a collection of vertex disjoint circuits $G_2$.  If $C$ is vertex disjoint from all the circuits in $G_2$, then simply adjoin $G_2 \cup C$ to the existing collection of circuits.
Otherwise $C$ will share vertices but not edges with (finitely many) circuits in $G_2$.  Travelling around $C$ we introduce loops through these existing (vertex disjoint) circuits to build a larger circuit, at each node of $C$ that is shared with (at most one) circuit of $G_2$.  The resulting enlarged circuit is then vertex disjoint from all circuits in $G_2$ that did not intersect $C$.
