Total Equitable Bondage Number of a Graph

If for any total dominating set D with ( ) v V G D   there exists a vertex u D  such that ( ) uv E G  and | ( ) ( ) | 1 d v d u   then D is called the total equitable dominating set. The minimum cardinality of the total equitable dominating set is called the total equitable domination number denoted by ( ) e t G  . The bondage number b(G) of a nonempty graph G is the minimum cardinality among all sets of edges 0 ( ) E E G  for which   0 ( ) G E G     . We introduced the concept of total equitable bondage number and proved several results.


Introduction
By a graph G we mean a simple, finite and undirected graph G with vertex set V(G) and edge set E(G).For any undefined term and notation in graph theory we rely upon West [13] while for standard terminology related to theory of domination we refer to Haynes et al. [4].
The set is either an element of D or is adjacent to an element of D. The minimum cardinality of a dominating set of G is called the domination number of G which is denoted by () G  . There are various domination models like global domination [7], total domination [2], equitable domination [8], equi-independent equitable domination [9], total near equitable domination in graphs [6], etc.
The concept of total domination was introduced by Cockayne et al. [2] while the concept of equitable domination was introduced by Swaminathan and Dharmalingam [8].Motivated through these concepts a new concept of total equitable domination was conceived by Basavanagoud et al. [1] and further explored by Vaidya and Parmar [10][11][12].The concepts of total dominating set and total equitable dominating set are useful for the formation of any committee.It is desirable that each committee member might feel comfortable knowing at least one member of the committee.In this situation total domination is useful while there is no difference of opinion between any two members or they differ on at most one issue.In this situation the concept of equitable domination is applicable.We have introduced the concept of total equitable bondage number.
is adjacent to at least one element in D. The minimum cardinality of total dominating set is called total domination number which is denoted by () We note that if G is a disconnected graph with n-components G 1 , G 2 , G 3 ,…,G n without isolated vertices and . Definition 1. 4 The bondage number b(G) of a nonempty graph G is the minimum cardinality among all sets of edges 0  .The concept of bondage number was introduced by Fink et al. [3] which is useful for measuring the vulnerability of the network under link failure.Many types of bondage numbers are introduced depending upon the characteristics of dominating sets.Some of them are total bondage number, roman bondage number, restrained bondage number are really noteworthy.
The concept of total bondage number was introduced by Kulli and Patwari [5].The total equitable bondage number is a bondage number with the additional property that removal of an edge subset from the given graph results in a graph with larger total equitable domination number.This concept is also useful for measuring the link failure of network.
bG of a graph G is the minimum cardinality among all sets of edges 0 ()

G E G
   .We avoid the removal of two consecutive edges in P n and C n as it give rise to isolated vertices in the resultant graphs which is not an ideal situation for the discussion of total equitable bondage number.

Definition 1.7
The wheel W n with n vertices is defined to be the join of K 1 and C n-1 .The vertex corresponding to K 1 is known as apex while the vertices corresponding to C n are known as rim vertices.We will state some of the existing results for our ready reference.Proposition 1.8 [1] For any path Proposition 1.11 [1] For any complete bipartite graph

Main Results
Theorem 2.1 The total equitable bondage number of path of order 3 n  is given by     Hence  

Conclusion
The concepts of bondage number and total bondage number were introduced by Fink et al. [3] and Kulli and Patwari [6] respectively, while we have introduced the concept of total equitable bondage number.Total equitable bondage numbers of some standard graph families have been investigated and determined the bounds for the total equitable bondage number of trees.
then D is called the total equitable dominating set.The minimum cardinality of the total equitable dominating set is called the total equitable domination number denoted by () e t G  .The bondage number b(G) of a nonempty graph G is the minimum cardinality among all sets of edges where d(u) denotes the degree of vertex u and d(v) denotes the degree of vertex v.The minimum cardinality of D is called the equitable domination number of G which is denoted by () If the graph H obtained by removal of any edges e of If u and v are not equitable vertices in V(T) then If there exists equitable vertices u and v in V(T) such that .This contradicts our assumption.Therefore all the edges of T are equitable edges.That is,