Abstract:

Let G = (V ,E) be a graph, for an element x ∈ V ∪E, the open total neighborhood
of x is denoted by Nt (x) = {y|y is adjacent to x or y is incident with x,y ∈
V ∪ E}, and Nt [x] = Nt (x) ∪ {x} is the closed one. A function f : V (G) ∪ E(G)→
{−1, 0, 1} is said to be a mixed minus domination function (TMDF) of G if
y∈Nt [x] f (y)  1 holds for all x ∈ V (G) ∪ E(G). The mixed minus domination
number γ  tm(G) of G is defined as
γ  tm(G) = min
 
x∈V ∪E
f (x)|f is a TMDF of G

.
In this paper, we obtain some lower bounds of the mixed minus domination number
of G and give the exact values of γ  tm(G) when G is a cycle or a path.

Key words: Mixed minus domination function , ｜ Mixed minus domination number