On Non-Zero Vertex Signed Domination

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For a graph G=(V,E) and a function f:V→{−1,+1}, if S⊆V then we write f(S)=∑v∈Sf(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 γsb(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 γsb(G) for several special classes of graphs is determined. Finally, we pose some open problems.