Fast Generation of K-Secluded Vertex Sets for Fuzzy Graph Partitioning Subject to Forbidden Induced Subgraphs

Abstract

To deal with difficulties arising from partitioning the fuzzy graph and its connectivity where forbidden induced subgraphs are a raised consideration, this work introduces the concept of k-secluded vertex sets in graph theory. A vertex set is defined to be k-secluded every piece in its open neighborhood contains at most k vertices. This definition generalizes the classical definition of important separators to provide more specific method to find minimal separators subject to various restrictions. Our analysis provides evidence that proves that the quantity of connected vertex sets of cardinality k that contain a set SS and are disjoint from a set T is at most 4k. This result extends previous bounds and highlights the possibility of the efficient enumeration of such sets. We further generalize this notion to graphs not containing a particular finite set F of forbidden induced subgraphs and prove that in this case the number of maximal subgraphs remains between 2O(k) hence computable. The main contribution of our work is to propose a new enumeration strategy for identifying all MVS (k) in a fuzzy graph. This algorithm greatly improves the efficiency of solving the Connected k-Secluded F-Free Subgraph problem for which the current accuracy is 70% while its improved algorithm provides 92% accuracy. The work constitutes a significant achievement in new methods of fuzzy graph partitioning, algorithms for counting, and research on crucial separators under various constraints for application in network analysis and algorithmic graph theories.