We prove that every connected subcubic graph G has two spanning trees \(T_1,T_2\) such that every component of \(G – E(T_1)\) is a path of length at most \(3\), and every component of \(G – E(T_2)\) is either a path of length at most \(2\) or a cycle.
Citation
Rui Li, Qing Cui. Spanning Trees in Subcubic Graphs[J], Ars Combinatoria, Volume 117. 411-415. .