Remarks on the minus (signed) total domination in graphs

Abstract

A function f : V (G) → {+1, 0, −1} defined on the vertices of a graph G is a minus total dominating function if the sum of its function values over any open neighborhood is at least one. The minus total domination number γ − t (G) of G is the minimum weight of a minus total dominating function on G. By simply changing "{+1, 0 − 1}" in the above definition to "{+1, −1}", we can define the signed total dominating function and the signed total domination number γ s t (G) of G. In this paper we present a sharp lower bound on the signed total domination number for a k-partite graph, which results in a short proof of a result due to Kang et al. on the minus total domination number for a k-partite graph. We also give sharp lower bounds on γ s t and γ − t for triangle-free graphs and characterize the extremal graphs achieving these bounds.

Dominating functions with integer values in graphs—a survey

“…Shan, et al [32] extended this result to k-partite graphs and characterized the extremal graphs. In fact, we can continue to generalize the result to K r+1 -free graphs where r 2 by applying the following well-known Turán theorem.…”

“…with equality if and only if r divides n. Theorem 11 [32] If G = (V, E) is a K r+1 -free graph of order n with δ(G) 1 and c = (δ(G) + 1)/2 , then…”

On the complexity of signed and minus total domination in graphs

“…Total domination in graphs has been extensively studied in the literature and surveyed in [5,6,8]. Recently many bounds on the signed and minus total domination numbers of graphs have been studied in [7,9,16,17,[19][20][21], but few papers studied the algorithmic complexity of them. From the algorithmic point of view, Harris and Hattingh [4] gave two linear-time algorithms to solve the signed and minus domination problems on trees, respectively.…”

Signed total Roman domination in graphs

“…The signed total domination number, denoted γ st (G), is the minimum weight of a signed total dominating function in G. Signed total domination in graphs is well studied in the literature, see for example, Henning (2004), Shan and Cheng (2008) and elsewhere.…”