Edge Monophonic Domination Number of Graphs

Arul Paul P Sudhahar ,Mohammed Abdul Khayyoom

doi:10.24297/jam.v11i10.812

Abstract

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 andÂ me(G) = q.Also there is a connected graph G such that (G) = p;me(G) = q and me(G) = p + q

Highlights

Example 2.1 Consider the graph G given in figure 01.Here M = {v4, v7, v8} is an edge monophonic set
By a graph G = (V, E) we consider a finite undirected graph without loops or multiple edges
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]

Summary

Introduction

Example 2.1 Consider the graph G given in figure 01.Here M = {v4, v7, v8} is an edge monophonic set . Theorem 2.2 :For any connected graph G of order n ,2 ≤ γm (G) ≤ γme G ≤ n Proof : Since a monophonic domination set need atleast two vertices, 2 ≤ γm (G) . Theorem2.3 :Each semi-extreme vertex of G belongs to every EMD set ofG.

Results

Conclusion