On vertex covers and matching number of trapezoid graphs

TL;DR

This paper improves algorithms for computing minimum vertex covers and related parameters in trapezoid graphs using binary indexed trees, reducing complexity from quadratic to near-linear time, and provides counterexamples for a previous matching algorithm.

Contribution

It introduces faster algorithms for vertex cover problems in trapezoid graphs and presents counterexamples for an existing maximum matching algorithm.

Findings

Algorithms for vertex covers now run in O(n log n) time.

Counterexamples demonstrate limitations of a previous O(n^2) matching algorithm.

Enhanced understanding of trapezoid graph properties and algorithms.

Abstract

The intersection graph of a collection of trapezoids with corner points lying on two parallel lines is called a trapezoid graph. Using binary indexed tree data structure, we improve algorithms for calculating the size and the number of minimum vertex covers (or independent sets), as well as the total number of vertex covers, and reduce the time complexity from $O (n^2)$ to $O (n \log n)$, where $n$ is the number of trapezoids. Furthermore, we present the family of counterexamples for recently proposed algorithm with time complexity $O (n^2)$ for calculating the maximum cardinality matching in trapezoid graphs.