An O(n^3) Time Algorithm for Recognizing Threshold Dimension 2 Graphs

Threshold dimension 2 graphs are the (edge-)intersection of two threshold graphs T1 and T2. Moreover they are the intersection graphs of points, axially parallel line segments and rectangles in the first quadrant of the Euclidean plane subject to the following constraints: 1. (1) line segments have one endpoint on one of the axes, 2. (2) the lower left corner of each rectangle is the origin and 3. (3) except for the above, every point, endpoint of a line segment and corner of a rectangle has unique x and y coordinates. Ma (1993) showed that the above geometrical representation called rectangle model can be constructed in O(n3) time providing the vertices that correspond to the rectangles are known. In this paper, we prove that there always exists a rectangle model in which the rectangles correspond to the vertices common to all maximum cliques. As the maximum cliques of a threshold dimension 2 graph can be found in O(n3), the overall running time of our recognition algorithm is O(n3), which compares favorably to the previous approaches with time complexity O(n5) and O(n4), respectively.