A fuzzy planar subgraph formation model for partitioning very large-scale integration networks

This study focuses on forming planar subgraphs in fuzzy graphs where the planarity value is not equal to 1. The paper commences by introducing the concepts of vertex-deletion (VD) and edge-deletion (ED) operations applied to fuzzy graphs. These operations aim to derive the fuzzy planar subgraphs and extensively examine the consequential significant findings. Additionally, maximal and maximum planar subgraphs in vertex-deletion (VD) and edge-deletion (ED) are presented alongside illustrative examples. A novel approach called the Planar Partition subgraph (PP subgraph) is introduced for identifying the fuzzy planar subgraphs from a fuzzy graph with an algorithm rooted in PP subgraphs for enhanced efficiency. Furthermore, the concept of thickness value in fuzzy graphs is constructed using PP-subgraphs is introduced. The discussion involves the thickness value of a fuzzy planar graph and its connection to the planarity value, supported by illustrative examples. Finally, an application made on PP subgraphs where the network needs to be partitioned into a finite number of subgraphs is demonstrated in a very large-scale integration network.