The Co-bondage (Bondage) number of fuzzy graphs and its properties

Abstract

In this paper, we define the Co-bondage number $b_c(G)$ and new type of non-bondage $\left\{b_{e n}\right.$ and $\left.b_{t n}\right\}$ for any fuzzy graph,and fuzzy strong line graph. A characterization is obtained for fuzzy strong line graphs $L_s(G)$ such that $L_s(G)$ is tree. A necessary condition for a fuzzy double strong line graph of cycle is a fuzzy trees and the exact value of $b_n(G)$ for any graph $G$ is found and exact values of $b_c, b_{e n}$ and $b_{t n}$ for some standard graphs are found and some bounds are obtained. Also,find the exact value of $b_{t n}(G)$ for any graph $G$ is found. Moreover we define neighbourhood extension also analysis it properties by using bondage arcs and we also obtained relationships between $b_c, b_{t n}(G)$ and $b_t$.