SOME NEWFANGLED RESULTS IN CONNECTED INTEGRITY IN GRAPHS

Abstract

            Facility Location Problem is a traditional optimization problem to decide the best location of a service spot in order to fulfill the needs of the entire vicinity. The optimal solution to this problem is reached by means of graph models. During the period of contagious diseases like COVID’19, the primary aim is to isolate the nodes of the network to a greater extend. Also while breaking the chain, one has the objective that the victim's region must be reduced as small as possible which in turn results in the less probability of spreading the disease. Both motives merge together in arranging medical camps in severely affected places. The foremost intention will be minimizing the cost of arranging medical camps and maximizing the easiness of medicine supply among them. The study on a graphical parameter named as Integrity yields an optimal solution to this problem. In order to share the availability of medicines in the camps, they must be inter connected. Motivated by this situation, the parameter namely, Connected Integrity, defined by  where the graph induced by S is connected and m(G - S) denotes the order of the  largest component of G - S is introduced. In this paper, we pay attention to study the properties of CI sets, realize graphs on CI values, generate CI values for trees and characterize trees with CI value 4. Also we introduce the derived graph namely z- Splitting graph.