Revisiting k-tuple Dominating Sets with Emphasis on Small Values of k

• Published in: Bulletin of the Malaysian Mathematical Sciences Society 2022-07, Vol.45 (4), p.1473-1487
• Main Authors: Samadi, Babak, Soltankhah, Nasrin, Mojdeh, Doost Ali
• Format: Article
• Language: English
• Subjects: Apexes; Applications of Mathematics; Graph theory; Graphs; Mathematics; Mathematics and Statistics; Upper bounds
• ISSN: 0126-6705; 2180-4206
• DOI: 10.1007/s40840-022-01269-1

For any graph G of order n with degree sequence d 1 ≥ ⋯ ≥ d n , we define the double Slater number s ℓ × 2 ( G ) as the smallest integer t such that t + d 1 + ⋯ + d t - e ≥ 2 n - p in which e and p are the number of end-vertices and penultimate vertices of G , respectively. We show that γ × 2 ( G ) ≥ s ℓ × 2 ( G ) , where γ × 2 ( G ) is the well-known double domination number of a graph G with no isolated vertices. We prove that the problem of deciding whether the equality holds for a given graph is NP-complete even when restricted to 4-partite graphs. We also prove that the problem of computing γ × 2 ( G ) is NP-hard even for comparability graphs of diameter two. Some results concerning these two parameters are given in this paper improving and generalizing some earlier results on double domination in graphs. We give an upper bound on the k -tuple domatic number of graphs with characterization of all graphs attaining the bound. Finally, we characterize the family of all full graphs, leading to a solution to an open problem given in a paper by Cockayne et al. (Networks 7: 247–261, 1977).