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A dominating set D of a graph G is called total if every vertex of V (G) is adjacent to at least one vertex of D, equivalently if N(D) = V (G) then D is called total dominating set. A dominating set D is called total equitable dominating set if it is total and for every vertex in V (G) − D there exists a vertex in D such that they are adjacent and difference between their degrees is at most one. The minimum cardinality of a total (total equitable) dominating set is called total (total equitable) domination number of G which is denoted by γ t (G)(γ e t (G)). We have investigated exact value of these parameters for some graphs.

If for any total dominating set D with ν ∈ V (G) − D there exists a vertex u ∈ D such that uν ∈ E (G) and |d(ν)−d(u)| ≤ 1 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 γet (G). The bondage number b(G) of a nonempty graph G is the minimum cardinality among all sets of edges E0 ⊆ E(G) for which γ(G – E0) > γ(G). We introduced the concept of total equitable bondage number and proved several results.

The bondage number of a nonempty graph is the minimum cardinality among all sets of edges for which . An equitable dominating set is called a total equitable dominating set if the induced subgraph has no isolated vertices. The total equitable domination number of is the minimum cardinality of a total equitable dominating set of . If and contains no isolated vertices then the total equitable bondage number of a graph is the minimum cardinality among all sets of edges for which . In the present work we prove some characterizations and investigate total equitable bondage number of Ladder and degree splitting of path.

A dominating set D is total if every vertex of V is adjacent to at least one vertex of D while D is called equitable if for every u in D there exists a vertex v in V D such that |d(u)d(v)| ≤ 1. A dominating set which is both total and equitable is called total equitable dominating set The total equitable domination number of G is the minimum cardinality of a total equitable dominating set of G which is denoted by γ e t (G). In this paper we contribute some general results on this concept.