A note on the tree decompositions of graphs

IN this note all graphs are undirected, finite and simple. For a subgraph H of G,ε(H) andμ(H) denote the number of edges in H and the number of cycles in H respectively. H[X]denotes the subgraph of H induced by X. Given two disjoint subsets X and Y of V(G), wewrite E(X, Y)={xy∈E(G)|x∈X, y∈Y}. Sometimes E(H, Y)=EG(V(H),Y) is used for a subgraph H of G-Y. If T is a tree of G and e=uv∈G-E(T)with{u,v}V(T), then T + e contains a unique cycle, denoted by C(T, e).A tree-decomposition {T, T, …, T} of a graph G is a partition of E (G), say,E(G)=EU EU…U E, such that for each i with 1≤i≤k, T=G[E] is a tree. We