Calculating the Centrality Values According to the Strengths of Entities Relative to their Neighbours and Designing a New Algorithm for the Solution of the Minimal Dominating Set Problem

Abstract

The dominating set problem in graph theory is an NP-complete problem for an arbitrary graph. There are many approximation-based studies in the literature to solve the dominating set problems for a given graph. Some of them are exact algorithms with exponential time complexities and some of them are based on approximation without robustness with respect to obtained solutions. In this study, the Malatya centrality value was used and a new Malatya centrality value was defined to solve the dominating set problem for a given graph. The improved algorithms have polynomial time and space complexities.

Keywords

• Dominating Set
• Malatya Centrality Value
• Graph Theory
• NP-Complete Problem

References

1. Bourgeois, N., Croce, F.D., Escoffier, B., Paschos, V.T.,” Fast algorithms for min independent dominating set”, Discrete Applied Mathematics, Vol:161, pp:558-572, 2013.
2. Goddard, W., Henning, M.A.,”Independent domination in graphs: A survey and recent results”, Discrete Mathematics, Vol:313, pp:839-854, 2013.
3. Grandoni, F.,”A note on the complexity of minimum dominating set”, Journal of Discrete Algorithms, Vol:4, pp:209-214, 2006.
4. Guha, S., Khuller, S.,”Approximation Algorithms for Connected Dominating Sets”, Algorithmica, Vol:20, pp:374-387, 1998.
5. Hagerup, T., “A strengthened analysis of an algorithm for Dominating Set in planar graphs”, Discrete Applied Mathematics, Vol:160, pp:793-798, 2012.
6. Karci, A., Yakut, S., Oztemiz, F.,” A New Approach Based on Centrality Value in Solving the Minimum Vertex Cover Problem: Malatya Centrality Algorithm”, Journal of Computer Science, Vol:7, pp:81-88,2022.
7. Khamis, S.M., Daoud, S.S., Essa, H.A.E,” A randomized algorithm for determining dominating sets in graphs of maximum degree five”, Theoretical Computer Science, Vol:410, pp:5122-5127, 2009.
8. Khuller, S., Yang, S.,” Revisiting Connected Dominating Sets: An Almost Optimal Local Information Algorithm”, Algorithmica, Vol:81, pp:2592-2605, 2019.

9. Van Rooij, J.M.M., Bodlaender, H.L.,” Exact algorithms for dominating set”, Discrete Applied Mathematics, Vol:159, pp:2147-2164, 2011.
10. Wawrzyniak, W.,”A local approximation algorithm for minimum dominating set problem in anonymous planar networks”, Distributed Computing, Vol:28, pp:321-331, 2015.