# The Edge Geodetic Domination Number

2520 Words Sep 11th, 2016 11 Pages
ABSTRACT In this paper the concept of upper edge geodetic domination number (UEGD number) and upper connected edge geodetic domination number (UCEGD number) of a graph is studied. An edge geodetic domination set (EGD set) S in a connected graph is minimal EGD set if no proper subset of S is an edge geodetic domination set. The maximum cardinality of all the minimal edge geodetic domination set is called UEGD number. An EGD set S in a connected graph is minimal CEGD set if no proper subset of S is a CEGD set. The maximum cardinality of all the minimal connected edge geodetic domination set is called UCEGD number. Here the UEGD number and UCEGD number of certain graphs are identified. Also for two positive integers p and q there exist some connected graph with EGD number p and UEGD number q. Similarly for two positive integers p and q there exist some connected graph with CEGD number p and UCEGD number q.
Keywords
Geodetic domination number, edge geodetic domination number, upper edge geodetic domination number, upper connected edge geodetic domination number.
AMS subject Classification: 05C12, 05C05

1 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 p and q respectively. For the basic graph theoretic notations and terminology we refer to Buckley and Harary [4].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. A…