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

R. Jahir Hussaina R. M. Karthik Keyanb,*

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

Authors :

R. Jahir Hussain^a R. M. Karthik Keyan^b,*

Author Address :

^a,bAssociate Professor of Mathematics, Jamal Mohamed College, Trichy,Tamil Nadu,India.

*Corresponding author.

Abstract :

In this paper, we define the Co-bondage number $b_{c}(G)$and new type of non-bondage{$b_{en}$ and $b_{tn}$} 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 $b_{c}$, $b_{en}$ and $b_{tn}$ for some standard graphs are found and some bounds are obtained. Also,find the exact value of $b_{tn}(G)$ 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 $b_{c}, b_{tn}(G)$ and $b_{t}$.

Keywords :

$gamma(G)$- Minimum dominating set ,$b_{c}(G)$- maximum co-bondage number ,b(G)- minimum bondage number , $b_{tn}$-maximum total non-bondage number, $b_{en}$- maximum efficient-bondage number ,$

DOI :

Article Info :

Received : March 12, 2017; Accepted : May 13, 2017.