Efficient algorithm for the vertex connectivity of trapezoid graphs

The intersection graph of a collection of trapezoids with corner points lying on two parallel lines is called a trapezoid graph. These graphs and their generalizations were applied in various fields, including modeling channel routing problems in VLSI design and identifying the optimal chain of non-overlapping fragments in bioinformatics. Using modified binary indexed tree data structure, we design an algorithm for calculating the vertex connectivity of trapezoid graph G with time complexity O(n log n), where n is the number of trapezoids. Furthermore, we establish sufficient and necessary condition for a trapezoid graph G to be bipartite and characterize trees that can be represented as trapezoid graphs.