Tree-width, path-width, and cutwidth

AbstractLet tw(G), pw(G), c(G), Δ(G) denote, respectively, the tree-width, path-width, cutwidth and the maximum degree of a graph G on n vertices. It is known that c(G≥tw(G). We prove that c(G)=O(tw(G)·Δ(G)·log n), and if ({Xi: i∈I}, T=(I,A)) is a tree decomposition of G with tree-width≤k then c(G)≤(k+1)·Δ(G)·c(T). In case that a tree decomposition is given, or that the tree-width is bounded by a constant, efficient algorithms for finding a numbering with cutwidth within the upper bounds are implicit in the proofs. We obtain the above results by showing that pw(G)=O(log n·tw(G)), and pw(G)≤(k+1)·c(T)