A lower bound on the total signed domination numbers of graphs

Abstract

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 .