Efficient algorithm for the vertex connectivity of trapezoid graphs

TL;DR

This paper presents an efficient $O(n \, \log n)$ algorithm for computing the vertex connectivity of trapezoid graphs using a modified binary indexed tree, and characterizes bipartite and tree trapezoid graphs.

Contribution

It introduces a novel $O(n \, \log n)$ algorithm for vertex connectivity of trapezoid graphs and provides conditions for bipartiteness and tree representation.

Findings

Vertex connectivity can be computed efficiently with the proposed algorithm.

Conditions for bipartiteness of trapezoid graphs are established.

Characterization of trees that are trapezoid graphs is provided.

Abstract

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.