ON THE SIGNED TOTAL DOMINATION NUMBER OF GENERALIZED PETERSEN GRAPHS P(n, 2)

Abstract

Abstract. Let G = (V,E) be a graph. A function f : V → {−1,+1}defined on the vertices of G is a signed total dominating function if thesum of its function values over any open neighborhood is at least one.The signed total domination number of G, γ st (G), is the minimum weightof a signed total dominating function of G. In this paper, we study thesigned total domination number of generalized Petersen graphs P(n,2)and prove that for any integer n ≥ 6, γ st (P(n,2)) = 2⌊ n3 ⌋ + 2t, wheret ≡ n(mod 3) and 0 ≤ t ≤ 2. 1. IntroductionFor notation and graph theory terminology we in general follow [5]. Spe-cially, let G be a graph with vertex set V (G) and edge set E(G). Let v be a ver-tex in V (G). The openneighborhood of v is N G (v) = {u ∈ V (G) | uv ∈ E(G)}.If the graph G is clear from the text, we simply write V , E and N(v) ratherthan V (G), E(G) and N G (v). If X ⊆ V , then hXi is the subgraph induced byX. The distance d(x,y) between two vertices x and y in G is the length of theshortest path from x to y. For a real-valued function f : V → R, the weight off is w(f) =P