Efficient Algorithms for Determining the Maximum Independent Sets in Graphs

Öz

It is known that there are many NP-hard and NP-complete problems in graph theory. The aim of this paper to solve the maximum independent set which is an NP-Hard and NP-Complete problem. In order to solve maximum independent set for given graph, a special spanning tree which is defined in this paper for the first time, is solved to obtain the fundamental cut-sets. The fundamental cut-sets are used to determine the effects of nodes. These effects of nodes are used to determine the elements of maximum independent set.

Anahtar Kelimeler

• Spanning Tree,Fundamental Cut-Sets,Efficient Algorithms,Effects of Nodes

Kaynakça

1. Brandstadt, A., Mosca, R., “Maximum weight independent set for Lclaw-free graphs in polynomial time”, Discrete Applied Mathematics, Vol:237, pp:57-64, 2018.
2. Jarden, A., Levit, V.E., Mandrescu, E.,”Critical and maximum independent sets of a graph”, Discrete Applied Mathematics, Vol: 247, pp:127-134, 2018.
3. Karci, A., Karci, Ş.,”Determination of Effective Nodes in Graphs”, International Conference on Science, Engineering & Technology, Mecca, Saudi Arabia, pp:25-28, 2020.
4. Karci, A., “Finding Innovative and Efficient Solutions to NP-Hard and NP-Complete Problems in Graph Theory”, Anatolian Science – Journal of Computer Science, 2020.
5. Laflamme, C., Aranda, A., Soukup, D.T., Woodrow, R.,”Balanced independent sets in graphs omitting large cliques”, Journal of Combinatorial Theory, Series B, Vol:137, pp:1-9, 2019.
6. Lin, M.-S.,”Counting independent sets and maximal independent sets in some subclasses of bipartite graphs”, Discrete Applied Mathematics, Vol:251, pp:236-244, 2018a.
7. Lin. M.-S., “Simple linear-time algorithms for counting independent sets in distance-hereditary graphs”, Discrete Applied Mathematics, Vol: 239, pp:144-153, 2018b.
8. Lin, M.-S., Chen, C.-M.,”Linear-time algorithms for counting independent sets in bipartite permutation graphs”, Information Processing Letters, Vol:122, pp:1-7, 2017.

9. Oh, S., “Maximal independent sets on a grid graph”, Discrete Mathematics, Vol:340, pp:2762-2768, 2017. Peramau, G., Perkins, W.,”Counting independent sets in cubic graphs of given girth”, Journal of Combinatorial Theory, Series B, Vol:133, pp:2018.
10. Sah, A., Sawhhney, M., Stoner, D., Zhao, Y.,”The number of independent sets in an irregular graph”, Journal of Combinatorial Theory, Series B, Vol:138, pp:172-195, 2019.
11. Wan, P., Tu, J., Zhang, S., Li, B., “Computing the numbers of independent sets and matchings of all sizes for graphs with bounded treewidth”, Applied Mathematics and Computation, Vol:332, pp:42-47, 2018.
12. Wan, P., Chen, X., Tu, J., Dehmer, M., Zhang, S., Emmert-Streib, F.,”On graph entropy measures based on the number of independent sets and matchings”, Information Sciences, Vol:516, pp:491-504, 2020.