Graph-Theoretic Algorithms Project Report Trapezoid Graphs

Trapezoid graphs are intersection graphs of trapezoids between two horizontal lines. They generalize both interval graphs and permutation graphs. Interval graphs are intersection graphs of intervals on the real line. Permutation graphs are intersection graphs of straight lines between two horizontal lines. They can be represented by a permutation diagram consisting of a pair of horizontal lines (a channel) where each line has n points. The points of one line are numbered from 1 to n and the points of the other line are numbered by a permutation of (1, 2, . . . , n). The graph has n vertices v1, v2, . . . , vn and vertices vi and vj are adjacent if and only if the line joining two i points intersect (cross) the line joining two j points. Figure 1 shows a permutation graph and a permutation diagram for it. Trapezoid graphs generalize these two graph classes by replacing points with intervals in the permutation diagram. The vertices vi and vj are adjacent if and only if the trapezoid joining the two intervals of i intersect the trapezoid joining the two intervals of j. In next section we define trapezoid graphs more precisely, present some equivalent characterization for them, and show an