Medium Domination Decomposition of Graphs

Abstract

A set of vertices  in a graph  dominates  if every vertex in  is either in  or adjacent to a vertex in . The size of any smallest dominating set is called domination number of . The concept of Medium Domination Number was introduced by Vargor and Dunder which finds the total number of vertices that dominate all pairs of vertices and evaluate the average of this value. The Medium domination Number is a notation which uses neighbourhood of each pair of vertices.  For G = (V, E) and ∀u,v∈ V if u, v are adjacent they dominate each other, then atleast dom (u, v) = 1. The total number of vertices that dominate every pair of vertices is defined as TDV(G)=∑dom(u,v), for every u,v∈V(G).  For any connected, undirected, loopless graph G of order p, the Medium Domination Number MD(G) = . In this paper we have introduced the new concept Medium Domination Decomposition. A decomposition of a graphG is said to be Medium Domination Decomposition (MDD) if