Properties of the Global Total k -Domination Number

Frank A. Hernández Mira, Ernesto Parra Inza, José M. Sigarreta Almira and Nodari Vakhania
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Frank A. Hernández Mira: Regional Development Sciences Center, Autonomous University of Guerrero, Los Pinos s/n, Suburb El Roble, Acapulco, Guerrero 39070, Mexico
Ernesto Parra Inza: Science Research Center, Autonomous University of Morelos, Cuernavaca 62209, Mexico
José M. Sigarreta Almira: Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero 39070, Mexico
Nodari Vakhania: Science Research Center, Autonomous University of Morelos, Cuernavaca 62209, Mexico

Mathematics, 2021, vol. 9, issue 5, 1-13

Abstract: A nonempty subset D ? V of vertices of a graph G = ( V , E ) is a dominating set if every vertex of this graph is adjacent to at least one vertex from this set except the vertices which belong to this set itself. D ? V is a total k -dominating set if there are at least k vertices in set D adjacent to every vertex v ? V , and it is a global total k -dominating set if D is a total k -dominating set of both G and G ¯ . The global total k -domination number of G , denoted by ? k t g ( G ) , is the minimum cardinality of a global total k -dominating set of G , GT k D-set. Here we derive upper and lower bounds of ? k t g ( G ) , and develop a method that generates a GT k D-set from a GT ( k ? 1 ) D-set for the successively increasing values of k . Based on this method, we establish a relationship between ? ( k ? 1 ) t g ( G ) and ? k t g ( G ) , which, in turn, provides another upper bound on ? k t g ( G ) .

Keywords: global total domination; total k -domination number (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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