On domination in the total torsion element graph of a module

Let R be a commutative ring with non-zero unity and M be a unitary R-module. Let T (M) be the set of torsion elements of M . Atani and Habibi [6] introduced the total torsion element graph of M over R as an undirected graph T (Γ(M)) with vertex set as M and any two distinct vertices x and y are adjacent if and only if x + y ∈ T (M). The main objective of this paper is to study the domination properties of the graph T (Γ(M)). The domination number of T (Γ(M)) and its induced subgraphs Tor(Γ(M)) and Tof(Γ(M)) has been determined. Some domination parameters of T (Γ(M)) are also studied. In particular, the bondage number of T (Γ(M)) has been determined. Finally, it has been proved that T (Γ(M)) is excellent, domatically full and well covered under certain conditions.


Introduction
The study of graphs associated to algebraic structures has become an exciting research topic in the last two decades, leading to many fascinating results and questions. Many fundamental papers assigning graphs to rings and modules have appeared recently, for instance see, [1,2,4,7,19]. In 2008, Anderson and Badawi [3] have introduced the total graph of a commutative ring and later on this notion has been generalised to many algebraic structures, in particular to module over a commutative ring (see [10,11]).
The concepts of dominating sets and domination numbers play a vital role in graph theory. Dominating sets are the focus of many books of graph theory, for example see [14] and [15]. But not much research has been done on the domination parameters of graphs associated to algebraic structures such as groups, rings, modules in terms of algebraic properties. However, some works on domination of graphs associated to rings and modules have appeared recently, for instance see, [9,12,17,18,20,21].
The study of the torsion elements is one of the important aspects of module theory. Atani and Habibi [6] have generalised the notion of total graph introduced by Anderson and Badawi [3] by introducing the total torsion element graph of a module M over a commutative R, denoted by T (Γ(M )), to be an undirected graph with all elements of M as vertices, and for distinct x, y ∈ M , the vertices x and y are adjacent if and only if x+y ∈ T (M ). Let T or(Γ(M )) be the (induced) subgraph of T (Γ(M )), with vertices T (M ), and let T of(Γ(M )) be the (induced) subgraph of T (Γ(M )) with vertices T of(M ). They have studied the characteristics of T (Γ(M )) and its two induced subgraphs T or(Γ(M )) and T of(Γ(M )) by considering two cases, T (M ) is a submodule of M or is not a submodule of M .
In this paper an attempt has been made to study the domination properties of the graph T (Γ(M )). The domination number of T (Γ(M )) and its induced subgraphs T or(Γ(M )) and T of(Γ(M )) has been determined. Some domination parameters of T (Γ(M )) has been studied. The bondage number of T (Γ(M )) has also been determined. Finally, it has been proved that T (Γ(M )) is excellent, domatically full and well covered under certain On domination in the total torsion element graph of a module 797 conditions.

Preliminaries
In this section, we recall the definitions,concepts and results which is needed in the later sections.
Throughout this paper, R is a commutative ring with non-zero unity and M is an unitary R-module, unless otherwise specified. An element a of a commutative ring R is called a zero-divisor of R if ab = 0 for some non-zero element b of R. Let Z(R) be the set of zero-divisors of R. Let For any undefined terminology in rings and modules we refer to [5,16].
By a graph G, we mean a simple undirected graph without loops. For a graph G, we denote by V (G) and E(G) the set of all vertices and edges respectively. We recall that a graph is finite if both V (G) and E(G) are finite sets, and we use the symbol |G| to denote the number of vertices in the graph G. We say that G is a null graph if E(G) = φ. A subgraph of G is a graph having all of its vertices and edges in G. A spanning subgraph of G contains all vertices of it. For any set S of vertices of G, the induced subgraph < S > is the maximal subgraph of G with vertex set S. Two vertices x and y of a graph G are connected if there is a path in G connecting them. Also, a graph G is connected if there is a path between any two distinct vertices. A graph G is disconnected if it is not connected. A graph G is complete if any two distinct vertices are adjacent. We denote the complete graph on n vertices by K n . A complete subgraph of G is called a clique. A maximum clique of G is a clique with largest number of vertices. The number of vertices in a maximum clique of G is called the clique number of G and it is denoted by ω(G). If the vertex set V (G) of the graph G are partitioned into two non-empty disjoint sets X and Y of cardinality |X| = m and |Y | = n, and two vertices are adjacent if and only if they are not in the same partite set, then G is called a bipartite graph. A graph G is called a complete bipartite graph if every vertex in X is connected to every vertex in Y . We denote the complete bipartite graph on m and n vertices by K m,n . For vertices x, y ∈ G one defines the distance d(x, y), as the length of the shortest path between x and y, if the vertices x, y ∈ G are connected and d(x, y) = ∞, if they are not. Then, the diameter of the graph G is The cycle is a closed path which begins and ends in the same vertex. The cycle of n vertices is denoted by C n . The girth of the graph G,denoted by gr(G) is the length of the shortest cycle in G and gr(G) = ∞ if G has no cycles.
For a subset S ⊆ V , < S > denotes the subgraph of G induced by S.
) and u is adjacent to v. The domination number γ(G) of G is defined to be minimum cardinality of a dominating set in G and such a dominating set is called γ-set of G. If G is a trivial graph, then γ(G) = 0. In a similar way, we define the strong domination number γ s and the weak domination number γ w . A graph G is called excellent if for every vertex v ∈ V , there exists a γ-set S containing v. A domatic partition of G is a partition of V into dominating sets in G. The maximum number of classes of a domatic partition of G is called the domatic number of G and is denoted by d(G). A graph G is called domatically full if d(G) = δ(G) + 1, which is the maximum possible order of a domatic partition of V (G) and δ(G) is the minimum degree of a vertex of G. The disjoint domination number γγ(G) defined by γγ(G) =min{|S 1 | + |S 2 | : S 1 , S 2 are disjoint dominating sets of G}. Similarly, we can define ii(G) and γi(G). The double domination parameters are referred to [13]. The bondage number b(G) is the minimum number of edges whose removal increases the domination number. A set of vertices S ⊆ V is said to be independent if no two vertices in S are adjacent in G. The independence number β 0 (G), is the maximum cardinality of an independent set in G. The minimum cardinality i(G) of a maximal independent set of a graph G is called the independent domination number of G. A graph G is called well-covered if β 0 (G) = i(G). For basic definitions and results in domination we refer to [14] and for any undefined graph-theoretic terminology we refer to [8].
Now we summarize some results on domination number and bondage number of a graph which will be useful for the later sections. (ii) γ(K n ) = 1 for a complete graph K n , but the converse is not true, in general and γ(K n ) = n for a null graph K n .
(vi) Domination number of a bistar graph is 2; because the set consisting of two centres of the graph is a minimal dominating set.
(vii) Let C n and P n be a n-cycle and a path with n vertices, respectively.
(ii) b(K n ) = n − 1 for a complete graph K n , but the converse is not true, in general and b(K n ) = 0 for a null graph K n .
(v) Let C n and P n be a n-cycle and a path with n vertices, respectively. Then b(P n ) = 1 and b(C n ) = 2.

Domination number of T (Γ(M)) and induced subgraphs
In this section, an attempt has been made to study the domination properties of the graph T (Γ(M )). In particular, the domination number of T (Γ(M )) and its induced subgraphs T or(Γ(M )) and T of(Γ(M )) have been determined.
The following proposition is an immediate observation of the above theorems.

Proposition 3.7:
Let M be a module over a commutative ring R such that T (M ) is a submodule of M . Then the following hold:     Proof. This is obvious as M being torsion, we have T (M ) = M and the result follows from proposition 3.7(1).
(2) If 2 = 1 R + 1 R / ∈ Z(R) then T of(Γ(M )) is the union of β − 1 2 disjoint K α,α 's.  On domination in the total torsion element graph of a module 803 (2) If R = Z 9 , then T (Γ(M )) is a disjoint union of one complete graph K 9 and 4 bipartite graphs K 9,9 . Proof. Let us consider the following two cases for Z(R).

Proposition 3.15:
Let M be a module over a commutative ring R such that T (M ) is a submodule of M . Then the following are equivalent: (1) γ(T (Γ(M ))) = 2.

Bondage number of T (Γ(M))
In this section, the bondage number of the graph T (Γ(M )) has been studied. We begin with the following proposition.

When is T (Γ(M)) is excellent, domatically full and well covered ?
In this section, some domination parameters of T (Γ(M )) has been studied. It has been proved that T (Γ(M )) is excellent, domatically full and well covered under some conditions.
We begin with the following proposition. Proof. If part follows directly from proposition 3.11 as γ(T (Γ(M ))) = β. Conversely, let S be a γ-set of T (Γ(M )). Let us assume that there exist j, k ∈ {1, 2, ..., β} such that x j ∈ x k + T (M ). Since |S| = β, so there exist a On domination in the total torsion element graph of a module 809 coset x + T (M ) such that x i / ∈ x + T (M ) for all x i ∈ S. Now, the vertices in −x + T (M ) cannot be dominated by S, a contradiction. (2) the domatic number d(T (Γ(M ))) = α.

Acknowledgement
The author would like to express his deep gratitude to the referee for a very careful reading of the article, and many valuable suggestions to improve the article.