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

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...

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Autores principales:	John,J., Flower,V. Sujin
Lenguaje:	English
Publicado:	Universidad Católica del Norte, Departamento de Matemáticas 2021
Materias:	Edge-to-edge geodetic domination number Edge-to-edge geodetic number Edge domination number Domination number Geodetic number
Acceso en línea:	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172021000300635
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spelling	oai:scielo:S0716-091720210003006352021-06-07The edge-to-edge geodetic domination number of a graphJohn,J.Flower,V. Sujin Edge-to-edge geodetic domination number Edge-to-edge geodetic number Edge domination number Domination number Geodetic number 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.info:eu-repo/semantics/openAccessUniversidad Católica del Norte, Departamento de MatemáticasProyecciones (Antofagasta) v.40 n.3 20212021-06-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172021000300635en10.22199/issn.0717-6279-4057
institution	Scielo Chile
collection	Scielo Chile
language	English
topic	Edge-to-edge geodetic domination number Edge-to-edge geodetic number Edge domination number Domination number Geodetic number
spellingShingle	Edge-to-edge geodetic domination number Edge-to-edge geodetic number Edge domination number Domination number Geodetic number John,J. Flower,V. Sujin The edge-to-edge geodetic domination number of a graph
description	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.
author	John,J. Flower,V. Sujin
author_facet	John,J. Flower,V. Sujin
author_sort	John,J.
title	The edge-to-edge geodetic domination number of a graph
title_short	The edge-to-edge geodetic domination number of a graph
title_full	The edge-to-edge geodetic domination number of a graph
title_fullStr	The edge-to-edge geodetic domination number of a graph
title_full_unstemmed	The edge-to-edge geodetic domination number of a graph
title_sort	edge-to-edge geodetic domination number of a graph
publisher	Universidad Católica del Norte, Departamento de Matemáticas
publishDate	2021
url	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172021000300635
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The edge-to-edge geodetic domination number of a graph

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