Efficient Algorithm for Dominatig Set In Graph Theory Based on Fundamental Cut-Set

Abstract

Determining the minimum dominating set in connected graphs is one of the most difficult problems defined as NP-hard. In this problem, it is aimed to determine the important nodes that can influence all nodes via the minimum number of nodes on the graph. In this study, an efficient near optimal algorithm showing a deterministic approach has been developed different from the approximation algorithms mentioned in literature for discovering dominating set. The algorithm has O(n3) time complexity in determining the Dominating Set (DS). At the same time, the algorithm is a original algorithm whose solution is not random by using fundamental cut-set. The DS algorithm consists of 3 basic phases. In the first phase of the algorithm, the algorithm that constructs the special spanning tree (Karci Max tree) of the graph is developed. In the second phase, the algorithm that finds the fundamental cut sets using the Kmax spanning tree is developed. In the last phase, Karci centrality node values are calculated with fundamental cut set and by using these Karci centrality node values, an algorithm has been developed to identify DS nodes. As a result of these three phases, the dominance values of the nodes on the graph and the DS nodes are calculated. The detected Karci centrality node values give priority to the node selection for determining the DS. All phases of the developed DS and Efficient node algorithms were coded in R programming language and the results were examined by running on sample graphs.