Edge Monophonic Domination Number of Graphs

In this paper the concept of edge monophonic domination num-ber of a graph is introduced.A set of vertices D of a graph G is edge mono-phonic domination set (EMD set) if it is both edge monophonic set and adomination set of G.The edge monophonic domination number (EMD num- ber) of G, me(G) is the cardinality of a minimum EMD set. EMD number of some connected graphs are realized.Connected graphs of order n with EMD number n are characterised.It is shown that for any two integers p and q such that 2 p q there exist a connected graph G with m(G) = p and me(G) = q.Also there is a connected graph G such that (G) = p;me(G) = q and me(G) = p + q


INTRODUCTION
By a graph G = (V, E) we consider a finite undirected graph without loops or multiple edges. The order and size of a graph are denoted by m and n respectively. For the basic graph theoretic notations and terminology we refer to Buckley and Harary [1]. For vertices u and v in a connected graph G, the distance d(u, v) is the length of a shortest u − v path in G. An u − v path of length d(u, v) is called u − v geodesic. A chord of a path P: u1 ,u2…un is an edge u i u j with j ≥ i + 2. An u − v path is monophonic path if it is chord less path. A monophonic set of G is a set M ⊂ V (G) such that every vertex of G is contained in a monophonic path of some pair of vertices of M.The monophonic number of a graph G is explained in [4] and further studied in [2]and [3]. The neighbourhood of a vertex v is the set N(v) consisting of all vertices which are adjacent with v. A vertex v is an extreme vertex if the sub graph induced by its neighbourhood is complete.
A graph G is said to be semi − extreme graph if every vertex of G is a semi extreme vertex. Every extreme vertex is a semiextreme vertex. Converse need not be true (see remark 2.2 in [2]).An acyclic connected graph is called tree [1]. A dominating set in a graph G is a subset of vertices of G such that every vertex outside the subset has a neighbour in it. The size of a minimum dominating set in a graph G is called the domination number of G and is denoted γ(G).A monophonic domination set of G is a sub set of V(G) which is both monophonic and dominating set of G.The minimum cardinality of a monophonic domonation set is denoted by γ m e G . A detailed study of monophonic domination set is available in [3]. An edge monophonic set of G is a subset M ⊂ V (G) such that every edge of G is contained in a monophonic path joining some vertices of M. The minimum cardinality among all the edge monophonic sets of G is called edge monophonic number and is denoted by me(G) A vertex v is an universal vertex of a graph G ifdeg(v) = n − 1. Edge monophonic set of a connected graph is studied in [2]. Remark 1.1:LetG be a connected graph of order n ≥ 3. If G contains exactly oneuniversal vertex, then me(G) = n − 1 (see [2]).  Theorem2.3 :Each semi-extreme vertex of G belongs to every EMD set ofG. Proof :LetD be an EMD set of G. Let u be a semi-extreme vertex ofG. Take u ∉ D. Let v be a vertex of < ( ) > such

BASIC CONCEPTS AND DEFINITIONS
Since D is also an edge monophonic set of G, the edge vu lies on the monophonic path P ∶ w, w1, …vi,u, v, vj …t, where w, t ∈ D. Since u is a semi extreme vertex of G, v andvj are adjacent in G and so P is not a monophonicpath of G.This contradicts our assumption. Proof:Consider the following cases.  Therefor γ m e H = p. Now A is not an EMD set since the edge u2u4 isnot lies in any edge monophonic path. But A ∪ {u2} is an EMD set. Therefor γ m e H = p − 1 + 1 + 1 = p + 1 = q. Case 3: Let p = 2, q ≥ 4 . Consider the graph K given in figure 06 .Kis obtained using the path P ∶ u, v, w of three vertices, by adding q -2 new vertices x1,x2…xq-2 and join these vertices with u, v, w.
Here v is auniversal vertex. Therefore γ m e K = │K│ = q − 2 + 3 − 1 = q. But A = u, w is a monophonicdomination set of K.Therefore the edge monophonic domination number γ m e K = 2.
Case 5 : Let p = q.Take T as the bipartrite graph K1,p. Thenγ m L = γ m e L = p.