# On the upper geodetic global domination number of a graph

## DOI:

https://doi.org/10.22199/issn.0717-6279-2020-06-0097## Keywords:

Geodetic set, Dominating set, Geodetic domination, Geodetic global domination, Upper geodetic global domination number## Abstract

A set S of vertices in a connected graph G = (V, E) is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A set D of vertices in G is called a dominating set of G if every vertex not in D has at least one neighbor in D. A set D is called a global dominating set in G if S is a dominating set of both G and ?. A set S is called a geodetic global dominating set of G if S is both geodetic and global dominating set of G. A geodetic global dominating set S in G is called a minimal geodetic global dominating set if no proper subset of S is itself a geodetic global dominating set in G. The maximum cardinality of a minimal geodetic global dominating set in G is the upper geodetic global domination number ?g+(G) of G. In this paper, the upper geodetic global domination number of certain connected graphs are determined and some of the general properties are studied. It is proved that for all positive integers a, b, p where 3 ? a ? b < p, there exists a connected graph G such that ?g(G) = a, ?g+(G) = b and |V (G)| = p.## References

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## Published

## How to Cite

*Proyecciones (Antofagasta, On line)*, vol. 39, no. 6, pp. 1627-1646, Nov. 2020.

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Copyright (c) 2020 X. Lenin Xaviour, S. Robinson Chellathurai

This work is licensed under a Creative Commons Attribution 4.0 International License.