Circular trapezoid graphs and circle trapezoid graphs: characteristics and algorithms

Abstract

Along with the direction that generalizes interval graphs and permutation graphs to (subclasses of) trapezoid graphs, researchers are now trying to generalize the class of trapezoid graphs. A circle trapezoid is the region in a circle that lies between two non-crossing chords, and the circle trapezoid graphs are the inter-section graphs of circle trapezoids in a circle;note that circle trapezoid graphs properly contains trapezoid graphs, circle graphs and circular-arc graphs as subclasses. Circle trapezoid graphs shall not be confused with circular trapezoid graphs. Here a circular trapezoid is the region within two parallel circles that lies between two non-crossing segments. It follows that the circular trapezoid graphs are the intersection graphs of circular trapezoids between two parallel circles. In this paper, the author presents results on two superclasses of trapezoid graphs including circle trapezoid graphs and circular trapezoid graphs. We show that circle trapezoid graphs and circular trapezoid graphs are two distinct classes of graphs;further, we show that the maximum weighted independent set on circular trapezoid graphs can be found in O(n²loglogn)time , and the minimum weighted independent dominating set of circular trapezoid graphs can be found in O(n²logn)time