On the structure of trapezoid graphs Original Research Article

Consider two parallel lines each containing n intervals, labelled 1 to n, where two intervals with the same label define a trapezoid with that label. The intersection graph of such a set of trapezoids is called a trapezoid graph. Trapezoid graphs are perfect and strictly contain both interval graphs and permutation graphs. In this paper we study the structure of trapezoid graphs and show that an operation called vertex splitting allows a trapezoid graph to be transformed into a permutation graph with special properties. Vertex splitting achieves this transformation by replacing the trapezoid representing a certain vertex v by its two end lines joining the ends of the intervals. These two lines now represent vertices v1 and v2. Continuing this operation eventually yields a special permutation graph. As a corollary of these results we get an O(n3) algorithm for recognizing a trapezoid graph and constructing a trapezoid representation of it. Although other trapezoid recognition algorithms are faster, ours is entirely graph theoretical and is easily implemented.