We define the nodes of this tree T to be the set I, and its edges to be the set F. The subset of V associated with node i of T is denoted by X i .
a tree decomposition is therefore defined as a pair ({X i | i ∈ I}, T = (I, F)), where each X i , i ∈ I, is a subset of V, and tree T has the following properties r Vertex cover: The nodes of the tree T cover all the vertices of G: r Edge cover: Each edge in G is included in some node of T. for all r Running intersection: The nodes of T containing a given vertex of G form a connected subtree.