Treewidth 15.1 Review 15.2 Fixed-parameter Tractability of Sat

Definition 1 A tree decomposition of a graph G(V,E) is a pair (T, χ) where T = (I, F ) is a tree and χ = {χi|i ∈ I} is a family of subsets of V such that (a) ⋃ i∈I χi = V , (b) for each edge e = {u, v} ∈ E, there exists i ∈ I such that both u and v belong to χi, and (c) for all v ∈ V , there is a set of nodes {i ∈ I|v ∈ χi} that forms a connected subtree of T . The width of a tree decomposition (T, χ) is given by maxi∈I(|χi| − 1).