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, Secon d, these graphs are exactly the pathwidth obstructions of order w + 3. Finally, although there is only one obstruction of order w + 2 for wi dth w, the number of obstructions of order w + 3 is bounded below by a n exponential function of root w.