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

• John, J. ^[2] ; Sujin Flower, V. ^[1]
  1. [1] Holy Cross College

     Holy Cross College

     Township of Portage, Estados Unidos

     
  2. [2] Government College of Engineering
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 40, Nº. 3, 2021 (Ejemplar dedicado a: In progress (June 2021). This issue is in progress. Contains articles that are final and fully citable.), págs. 635-658
• Idioma: inglés
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• Resumen

  • Let G = (V, E) be a connected graph with at least three vertices. A set S Í E 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 ¡gee(G) of G is the minimum cardinality of its edge-to-edge geodetic dominating sets and any edge-to-edge geodetic dominating set of minimum cardinality is said to be a gee- set of G. Some general properties satisfied by this concept are studied. Connected graphs of size m?2 with edge-to-geodetic domination number 2 or m or m-1 are charaterized. We proved that if G is a connected graph of size m ? 3 and G­ is also connected,then 4 ?¡gee(G) + ¡gee(G­) ? 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 and b with 2 ?a ? b, there exists a connected graph G with gee(G) = a and ¡gee(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 ¡e(G) = a and¡ gee(G) = b, where ¡e(G) is the edge domination number of G and gee(G) is the edge-to-edge geodetic number of G.