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.Comment: 12 pages, 2 figure

On the variable common due date, minimal tardy jobs bicriteria two-machine flow shop problem with ordered machines

We consider a special case of the ordinary NP-hard two-machine flow shop problem with the objective of determining simultaneously a minimal common due date and the minimal number of tardy jobs. In [S. S. Panwalkar, C. Koulamas, An O(n^2) algorithm for the variable common due date, minimal tardy jobs bicriteria two-machine flow shop problem with ordered machines, European Journal of Operational Research 221 (2012), 7-13.], the authors presented quadratic algorithm for the problem when each job has its smaller processing time on the first machine. In this note, we improve the running time of the algorithm to O(n log n) by efficient implementation using recently introduced modified binary tree data structure.Comment: 6 pages, 1 algorith