The structure and number of obstructions to treewidth

For each pair of nonadjacent vertices in a graph, consider the greater of the degrees of the two vertices. The minimum of these maxima is a lower bound on the treewidth of a graph, unless it is a complete graph. This bound has three consequences. First, the obstructions of order w + 3 for treewidth w have a simple structural characterization, Second, these graphs are exactly the pathwidth obstructions of order w + 3. Finally, although there is only one obstruction of order w + 2 for width w, the number of obstructions of order w + 3 is bounded below by an exponential function of root w.