THE MONOPHONIC GLOBAL DOMINATION NUMBER OF A GRAPH

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. A set M ⊆ V is said to be a monophonic global dominating set of G if M is both a monophonic set and a global dominating set of G. The minimum cardinality of a monophonic global dominating set of G is the monophonic global domination number of G and is denoted by γm(G). A monophonic global dominating set of cardinality γm(G) is called a γm-set of G. The monophonic global domination number of certain classes of graphs are determined. It is proved that 2≤ γm(G)≤ γg(G)≤ n, where γg(G) is a geodetic global domination number of a G. It is shown that for every pair of positive integers a and b with 2 ≤ a ≤ b, there exists a connected graph G such that γm(G) = a and γg(G) = b.


INTRODUCTION
By a graph G = (V, E), we mean a finite,undirected connected graph without loops or multiple edges. The order and size of G are denoted by m and n respectively. For basic graph theoretic terminology, we refer to [2]. Two vertices u and v are said to be adjacent if uv is an edge of G. If uv ∈ E(G), we say that u is a neighbor of v and denote by N(v), the set of The geodetic number g(G) of G is the minimum order of its geodetic sets and any geodetic set of order g(G) is a geodetic basis or a g-set of G. The geodetic number of a graph was studied in [1,3,5,[20][21][22][23]26]. A chord of a path P is an edge which connects two non-adjacent vertices of P.
The domination number, γ(G), of a graph G denotes the minimum cardinality of such dominating sets of G. A minimum dominating set of a graph G is hence often called as a γ-set of G. The domination concept was studied in [7]. A subset D ⊆ V is called a global dominating set in G if D is a dominating set of both G and G. The global domination number γ(G) is the minimum cardinality of a minimal global dominating set in G. The concept of global domination in graph was introduced in [25,29,30]. A set S ⊆ V is said to be a geodetic global dominating set of G if S is both a geodetic set and a global dominating set of G. The minimum cardinality of a geodetic global dominating set of G is the geodetic global domination number of G and is denoted by γ g (G). A geodetic global dominating set of cardinality γ g (G) is called a γ g -set of G. The concept of geodetic global domination in graph was studied in [4,24]. The domination concepts is used in networks. The geodetic and monophonic concepts are used in social networks. By applying the monophonic (geodetic) global domination concepts, there is a effectiveness in the networks. Throughout the following G denotes a connected graph at least two vertices. The following theorem is used in the sequel.  Example 2.2. For the graph G given in Figure 2  (ii)Each extreme vertex of a connected graph belong to every monophonic global dominating set of G.

THE MONOPHONIC GLOBAL DOMINATION NUMBER OF A GRAPH
(iii)Each universal vertex of a connected graph belong to every monophonic global dominating set of G.
In the following we determine the monophonic global dominating number of some standard graphs.
Proof. This follows from Observation 2.4(ii) and (iii).
Proof. If n = 2 or 3, then the result follows from Theorem 2.5 and 2.6. So, let n ≥ 4. Consider we can prove that γ m (G) = n + 2 3 .
For n ≥ 6, we consider three cases.
So, let n ≥ 6. By Observation 2.4(iii), x belongs to every monophonic global dominating set of  Proof. Let S be a geodetic global dominating set of G. Then S is a geodetic set and a global dominating set of G. Since every u-v geodesic is a u-v monophonic path, S is a monophonic set of G. Hence it follows that S is a global dominating set of G.
Let G be a connected graph of order n. Then 2 ≤ γ m (G) ≤ γ g (G) ≤ n.
Proof. This follows from Theorem 2.8. Proof. In a distance-hereditary graph, every u-v monophonic path is a u-v geodesic. Hence it follows that every monophonic global dominating set of G is a geodetic global dominating set of G. Therefore γ g (G) ≤ γ m (G). Hence the result follows from Corollary 3.2.
In the view of Corollary 3.2, we have the following realization result. Proof. For a = b, let G = K 1,a−1 . Then the result follows from Theorem 2.6. So, let 2 ≤ a < b.
Let P : x, y, z be a path on three vertices. Let P i :

CONCLUSION
In this article, we introduced the concept of the monophonic global domination number of a graph and studied some of its general properties. It can be further investigated to find out under which conditions the lower bound and the upper bound of the monophonic global domination number are sharp.

ACKNOWLEDGMENT
We are thankful to the referees form their constructive and detailed comments and suggestions which improved the paper overall.