In this paper, we consider two restricted types of dynamic orthogonal segment intersection search problems. One is the problem in which the underlying set is updated only by insertions. In the other, the set is updated only by deletions. We show that an intermixed sequence of O(n) queries and updates in both problems can be executed on-line in O(n log n+K) time and O(n log n) space, where K is the total number of reported intersections. Our algorithms utilize set-union and set-splitting algorithms. Especially, we present a linear-time algorithm for the incremental set-splitting problem, and use it for the problem in which only insertions are allowed. The data structures developed here give new better solutions for various geometric problems on orthogonal segments. We give a paradigm of solving those geometric problems by combining graph algorithms with these data structures, and describe a variety of applications.