Let G be a finite connected simple graph with a vertex set V (G) and an edge set E(G). A total signed domination function of G is a function f : V (G) ∪ E(G) → {?1, 1}. The weight of f is w(f) = Σ x∈V(G)∪E(G) f(x). For an element x ∈ V (G) ∪ E(G), we define $f[x] = \sum\nolimits_{y \in N_T [x]} {f(y)} $ . A total signed domination function of G is a function f : V (G) ∪ E(G) → {?1, 1} such that f[x] ? 1 for all x ∈ V (G) ∪ E(G). The total signed domination number γ s * (G) of G is the minimum weight of a total signed domination function on G.In this paper, we obtain some lower bounds for the total signed domination number of a graph G and compute the exact values of γ s * (G) when G is C n and P n .
Xin-zhong LU Department of Mathematics,Zhejiang Normal University,Jinhua 321004,China
