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Nordhaus-Gaddum bounds for upper total domination

Haynes Teresa W., Henning Michael A.

Opuscula Mathematica

2022

Vol. 42, no. 4

573--582

A set S of vertices in an isolate-free graph G is a total dominating set if every vertex in G is adjacent to a vertex in S. A total dominating set of G is minimal if it contains no total dominating set of G as a proper subset. The upper total domination number Γt(G) of G is the maximum cardinality of a minimal total dominating set in G. We establish Nordhaus-Gaddum bounds involving the upper total domination numbers of a graph G and its complement G. We prove that if G is a graph of order n such that both G and G are isolate-free, then Γt(G) + Γt(G) ≤ n + 2 and Γt(G)Γt(G) ≤ ¼ (n + 2)2, and these bounds are tight.

Remarks on the outer-independent double Italian domination number

Volkman Lutz

Opuscula Mathematica

2021

Vol. 41, no. 2

259--268

Let G be a graph with vertex set V(G). If u ∈ V(G), then N[u] is the closed neighborhood of u. An outer-independent double Italian dominating function (OIDIDF) on a graph G is a function ƒ : V(G) —> {0, 1, 2, 3} such that if ƒ (v) ∈ {0, 1} for a vertex v ∈ V(G), then [formula], and the set {u ∈ V(G) : ƒ (u) = 0} is independent. The weight of an OIDIDF ƒ is the sum [formula]. The outer-independent double Italian domination number [formula] equals the minimum weight of an OIDIDF on G. In this paper we present Nordhaus-Gaddum type bounds on the outer-independent double Italian domination number which improved corresponding results given in [F. Azvin, N. Jafari Rad, L. Volkmann, Bounds on the outer-independent double Italian domination number, Commun. Comb. Optim. 6 (2021), 123-136]. Furthermore, we determine the outer-independent double Italian domination number of some families of graphs.