  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.
