  The 3-decomposition conjecture is wide open. It asserts that every finite
connected cubic graph can be decomposed into a spanning tree, a disjoint union
of cycles, and a matching. We show that every such decomposition is derived
from a homeomorphically irreducible spanning tree (HIST). This allows us to
propose a novel reformulation of the 3-decomposition conjecture: the
HIST-extension conjecture.
  We also prove that the following graphs are reducible configurations with
respect to the 3-decomposition conjecture: the triangle, the K_{2,3}, the
Petersen graph with one vertex removed, the claw-square, the twin-house, and
the domino. As an application, we show that all 3-connected graphs of
tree-width at most 3 or of path-width at most 4 satisfy the 3-decomposition
conjecture and that a 3-connected minimum counterexample to the conjecture is
triangle-free, all cycles of length at most 6 are induced, and every edge is in
the centre of an induced P_6.
  Finally, we automate the naive part of the process of checking whether a
configuration is reducible and we prove that all graphs of order at most 20
satisfy the 3-decomposition conjecture.
