Tree decomposition

A tree decomposition of graphG = (V, E) is referred to as a partition of edge setE into edge-disjoint trees. Given (not necessarily distinct) vertices u₁, u₂ ⋯ u_k∈V with k⩾2, a sufficient and necessary condition is given for a connected graphG = (V, E) to have a tree decompositionT ₁,T ₂⋯T _k such thatV(T ₁)=V _i,i=1, 2, ⋯ k.