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The 3-decomposition conjecture is wide open. It asserts that every 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), which is surprising since a graph with a HIST trivially satisfies the conjecture.We also prove that the following graphs are reducible configurations with respect to the 3-decomposition conjecture: the Petersen graph with one vertex removed, the claw-square, the twin-house, and the domino. As an application, we show that all graphs 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 P6. Finally, employing a computer, we prove that all graphs of order at most 20 satisfy the 3-decomposition conjecture.

A decomposition of a graph is a set of subgraphs whose edges partition those of G. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph, and a matching. It has been settled for special classes of graphs, one of the first results being for Hamiltonian graphs. In the past two years several new results have been obtained, adding the classes of plane, claw-free, and 3-connected tree-width 3 graphs to the list.In this paper, we regard a natural extension of Hamiltonian graphs: removing a Hamiltonian cycle from a cubic graph leaves a perfect matching. Conversely, removing a perfect matching M from a cubic graph G leaves a disjoint union of cycles. Contracting these cycles yields a new graph G M . The graph G is star-like if G M is a star for some perfect matching M , making Hamiltonian graphs star-like. We extend the technique used to prove that Hamiltonian graphs satisfy the 3-decomposition conjecture to show that 3-connected star-like graphs satisfy it as well.

We present an interactive framework that, given a membership test for a graph class G and a number k ∈ N, finds and tests unavoidable sets for the class G k of graphs in G of path-width at most k. We put special emphasis on the case that G is the class of cubic graphs and tailor the algorithm to this case. In particular, we introduce the new concept of high-degree-first path-decompositions, which yields highly efficient pruning techniques.Using this framework we determine all extremal girth values of cubic graphs of path-width k for all k ∈ {3, . . . , 10}. Moreover, we determine all smallest graphs which take on these extremal girth values. As a further application of our framework we characterise the extremal cubic graphs of path-width 3 and girth 4.

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