The Upper Total Restrained Edge Geodetic Domination Number Of A Graph

Abstract

A setof vertices of a connected graphis a restrained geodetic dominatingset,if either oris a geodetic dominating set with the subgraphinduced byhas no isolated vertices. The minimum cardinality of a restrained geodetic dominating set of  is called the restrained geodetic domination number and is denoted byA total restrained edge geodetic dominating set S in a connected graph G is called a minimal total restrained edgegeodetic dominating set of G, if no proper subset of S is a total restrained edge geodetic dominating set of G . The upper total restrained edgegeodetic domination numberis the maximum cardinality of a minimal total restrained edge geodetic dominating set of G.The upper total restrained geodetic domination number of certain classes of graphs are determined.It is shown that for every pair of integersa, b  with, there exists a connected graph G of order bsuch that.  Also, for any five  integers a, b, c ,d  and ewith   there exists a connected graph Gsuch that ,  ,  ,and