Let Γ−t (G) be upper minus total domination number of G. In this paper, We establish an upper bound of the upper minus total domination number of a 6-regular graph G and characterize the extremal graphs attaining the bound. Thus, we partially answer an open problem by Yan, Yang and Shan. AMS Subject Classification: 05C69

A function f : V (G) → {+1,−1} defined on the vertices of a graph G is a signed dominating function if for any vertex v the sum of function values over its closed neighborhood is at least 1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. By simply changing “{+1,−1}” in the above definition to “{+1, 0,−1}”, we can define the minus dominating f...

3 A function f : V (G) → {+1,−1} defined on the vertices of a graph G is a signed domi4 nating function if for any vertex v the sum of function values over its closed neighborhood 5 is at least one. The signed domination number γs(G) of G is the minimum weight of a 6 signed dominating function on G. By simply changing “{+1,−1}” in the above definition 7 to “{+1, 0,−1}”, we can define the minus ...

A function f :V (G) → {−1, 0, 1} defined on the vertices of a graph G is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. An MTDF f is minimal if there does not exist an MTDF g:V (G) → {−1, 0, 1}, f = g, for which g(v) f (v) for every v ∈ V (G). The weight of an MTDF is the sum of its function values over all vertices. The mi...

We show that the total domination number of a simple connected graph is greater than the average distance of the graph minus one-half, and that this inequality is best possible. In addition, we show that the domination number of the graph is greater than two-thirds of the average distance minus one-third, and that this inequality is best possible. Although the latter inequality is a corollary t...

A three-valued function f defined on the vertices of a graph G = (V,E), f : V , ( 1 , 0 , 1), is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every v E V, f(N[v])>~ 1, where N[v] consists of v and every vertex adjacent to v. The weight of a minus dominating function is f ( V ) = ~ f (v) , over all vertices v E V. The mi...

In this paper we determine the exact values of the signed domination number, signed total domination number, and minus domination number of complete multipartite graphs, which substantially generalizes some previous results obtained for special subclasses of complete multipartite graphs such as cliques and complete bipartite graphs.