  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.
