(A) study on the modified 2 and 3-center in a tree network

Abstract

We introduce the formulation of a modified P-center. And efficient algorithms for finding the modified 2-center and modified 3-center of a tree are given. Critical vertices are defined to find the modified 2-center and 3-center. In the modified 2-center, we find an algorithm whose complexity is a linear function of the number of vertices. Also we find a linear time algorithm for the modified 2-center in continuous case. In the modified 3-center, we show that there are at most 4 equivalence classes over the critical vertices and using the fact, we find an $O(n^3)$ algorithm.