On Non-Zero Vertex Signed Domination.

For a graph  G = (V , E)  and a function  f : V → { − 1 , + 1 } , if  S ⊆ V  then we write  f (S) = ∑ v ∈ S f (v) . A function  f  is said to be a non-zero vertex signed dominating function (for short, NVSDF) of  G  if  f (N [ v ]) = 0  holds for every vertex  v  in  G , and the non-zero vertex signed domination number of  G  is defined as  γ s b (G) = max { f (V) | f   is   an   NVSDF   of   G }.  In this paper, the novel concept of the non-zero vertex signed domination for graphs is introduced. There is also a special symmetry concept in graphs. Some upper bounds of the non-zero vertex signed domination number of a graph are given. The exact value of  γ s b (G)  for several special classes of graphs is determined. Finally, we pose some open problems. [ABSTRACT FROM AUTHOR]