Abstract
AbstractIn this paper, we extend the findings of recent studies on k-rainbow total domination by placing our focus on its computational complexity aspects. We show that the problem of determining whether a graph has a 2-rainbow total dominating function of a given weight is NP-complete. This complexity result holds even when restricted to planar graphs. Along the way tight bounds for the k-rainbow total domination number of rooted product graphs are established. In addition, we obtain the closed formula for the k-rainbow total domination number of the corona product $$G*H$$
G
∗
H
, provided that H has enough vertices.
Publisher
Springer Science and Business Media LLC
Reference22 articles.
1. Berge, C.: Sur le couplage maximum d’un graphe. C. R. Acad. Sci. Paris 247, 258–259 (1958)
2. Brešar, B.: Rainbow domination in graphs. Series developments in mathematics. In: Haynes, T.W., Hedetniemi, S.T., Henning, M.A. (eds.) Topics in Domination in Graphs. Springer, Cham (2020)
3. Brešar, B., Kraner Šumenjak, T.: On the $$2$$-rainbow domination in graphs. Discrete Appl. Math. 155, 2394–2400 (2007)
4. Brešar, B., Henning, M.A., Rall, D.F.: Rainbow domination in graphs. Taiwanese J. Math. 12, 213–225 (2008)
5. Cabrera Martínez, A., Rodríguez-Velázquez, J.A.: Total domination in rooted product graphs. Symmetry 12(11), 1929 (2020). https://doi.org/10.3390/sym12111929