A note on the tree decompositions of graphs

<正> 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 EG（X, Y）={xy∈E（G）|x∈X, y∈Y}. Sometimes EG（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 {T1, T2, …, Tk} of a graph G is a partition of E （G）, say,E（G）=E1 U E2 U…U Ek, such that for each i with 1≤i≤k, Ti=G[Ei] is a tree. We