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Properties of the Global Total k -Domination Number

Author

Listed:
  • 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)

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 γ 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 γ 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 γ ( k − 1 ) t g ( G ) and γ k t g ( G ) , which, in turn, provides another upper bound on γ k t g ( G ) .

Suggested Citation

  • Frank A. Hernández Mira & Ernesto Parra Inza & José M. Sigarreta Almira & Nodari Vakhania, 2021. "Properties of the Global Total k -Domination Number," Mathematics, MDPI, vol. 9(5), pages 1-13, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:5:p:480-:d:506314
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