The edge-to-edge geodetic domination number of a graph

J John,V Sujin Flower

doi:10.22199/issn.0717-6279-4057

Abstract

Let G = (V, E) be a connected graph with at least three vertices. A set S ⊆ E(G) is called an edge-to-edge geodetic dominating set of G if S is both an edge-to-edge geodetic set of G and an edge dominating set of G. The edge-to-edge geodetic domination number γgee(G) of G is the minimum cardinality of its edge-to-edge geodetic dominating sets. Some general properties satisfied by this concept are studied. Connected graphs of size m with edge-to-edge geodetic domination number 2 or m or m − 1 are characterized. We proved that if G is a connected graph of size m ≥ 4 and Ḡ is also connected, then 4 ≤ γgee(G) + γgee(Ḡ) ≤ 2m − 2. Moreover we characterized graphs for which the lower and the upper bounds are sharp. It is shown that, for every pair of positive integers a, b with 2 ≤ a ≤ b, there exists a connected graph G with gee(G) = a and γgee(G) = b. Also it is shown that, for every pair of positive integers a and b with 2 < a ≤ b, there exists a connected graph G with γe(G) = a and γgee(G) = b, where γe(G) is the edge domination number of G and gee(G) is the edge-to-edge geodetic number of G.

Highlights

By a graph G = (V, E), we mean a finite, undirected connected graph without loops or multiple edges
The subgraph induced by set S of vertices of a graph G is denoted by hSi with V = S and E(hSi) = {uv ∈ E(G) : u, v ∈ S}
It is seen that an edge dominating set is not in general an edge-to-edge geodetic set in a graph G

Summary

Introduction

A set of vertices D in a graph G is called a dominating set of G if each vertex of V (G) − D is adjacent to some vertex of G. A set S ⊆ E(G) is called an edge-to-vertex geodetic set if every vertex of G is either incident with an edge of S or lies on a geodesic joining a pair of edges of S. The problem is to find the worst-case behavior of the network, i.e., the minimum number of routed calls when the network is saturated and no calls can be added For this we construct a bipartite graph G by connecting a line to a trunk if and only if the line can be switched to the trunk. In the case of designing the route for a shuttle, all the vertices are covered by the shuttle when considering edgeto-vertex geodetic sets, some of the edges may be left out. Throughout the following G denotes a connected graph with at least three vertices

The edge-to-edge geodetic domination number of a graph

Some results on the edge-to-edge geodetic domination number of a graph

Graph with small and large edge-to-edge geodetic domination number

Realization Results