Strongest Strong Cycles and $\theta$-Fuzzy Graphs

In this paper, new concepts are introduced to enhance the process of block identification in fuzzy graphs. Since reduction in the strength of the connectedness between two nodes occurs more frequently than total disconnection of a network, block identification in fuzzy graph networks is very important. A special type of cycle, which is called a strongest strong cycle, and a new connectivity parameter, which is called cycle connectivity, are introduced in fuzzy graphs. A subclass of fuzzy graphs known as θ-fuzzy graphs is identified, in which all the existing characterizations of blocks in graphs are true. A set of necessary conditions is obtained for a fuzzy graph to be a block, as well as a set of necessary and sufficient conditions for a θ-fuzzy graph to be a block.