Algorithms for closeness, additional closeness and residual closeness

Abstract

The residual and additional closeness are very important characteristics of graphs. They are measures of graphs’ vulnerability and growth potentials. Calculating the closeness, the residual, and the additional closeness of graphs is a difficult computational problem. In this article we propose an algorithm for additional closeness and an approximate algorithm for closeness. Calculating the residual closeness of graphs is the most difficult of the three closenesses. We use Branch and Bound like algorithms to solve this problem. In order for the algorithms to be effective, we need good upper bounds of the residual closeness. In this article we have calculated upper bounds for the residual closeness of 1-connected graphs. We use these bounds in combination with the approximate algorithm to calculate the residual closeness of 1connected graphs. We have done experiments with randomly generated graphs and have calculated the decrement in steps, delivered by the proposed algorithm.