On powers of m-trapezoid graphs

First a new class of graphs is introduced: m-trapezoid graphs are the intersection graphs of m-trapezoid, where an m-trapezoid is given by m + 1 intervals on m + 1 parallel lines. The main result of this paper is that if Gk − 1 is an m-trapezoid graph then Gk is also an m-trapezoid graph. This theorem has some interesting corollaries concerning interval graphs, trapezoid graphs and cocomparability graphs: If A is either of these classes, then Gk − 1 ϵ A implies Gk ϵ A. This answers an open question of P. Damaschke.