Triangle Graphs and Simple Trapezoid Graphs

In the literature of graph recognition algorithms, two of the most extensively covered graphs are interval graphs and permutation graphs [6, 2]. One way of generalizing both interval graphs and permutation graphs is the intersection graph of a collection of trapezoids with corner points lying on two parallel lines. Such a graph is called the trapezoid graph [3, 4]; Ma and Spinrad [8] show that trapezoid graphs can be recognized in O(n) time. On the other hand, Golumbic and Monma [7] introduce the class of tolerance graphs that also generalizes permutation graphs and interval graphs. Each vertex v ∈ V of a tolerance graph G = (V, E) corresponds an interval Iv on a line and a positive real number tv (the tolerance). Two vertices u, v are adjacent to each other, i.e., uv ∈ E iff |Iu ∩ Iv| ≥ min{tu, tv}. Furthermore, a tolerance graph is called bounded if tv ≤ |Iv| for all v ∈ V. Bogart et al. [1] showed that bounded tolerance graphs are actually parallelogram graphs; i.e., intersection graphs of parallelograms each of which has its horizontal lines on twoparallel lines. The most interesting thing shown in the study of these subclasses of tolerance graphs is that, while they are all subclasses of trapezoid graphs, we still do not know how to efficiently recognize these graphs. This gives us a motivation for trying to find other properties of these subclasses of trapezoid graphs, which are still superclasses of permutation and interval graphs. In this paper, the author presents results on two subclasses of trapezoid graphs including simple trapezoid graphs and triangle graphs. Here the intersection graph of rectangles and line segments whose two ends lie on two parallel lines is called the simple