Singed Total Domatic Number of a Graph

Abstract

Let G be a flnite and simple graph with vertex set V (G), k ‚ 1 an integer and let f : V (G) ! fik;k i1;¢¢¢ ;i1;1;¢¢¢ ;k i1;kg be 2k valued function. If P x2N(v) f(x) ‚ k for each v 2 V (G), where N(v) is the open neighborhood of v, then f is a Smarandachely k-Signed total dominating function on G. A set ff1;f2;:::;fdg of Smarandachely k-Signed total dominating function on G with the property that d P i=1 f i(x) • k for each x 2 V (G) is called a Smarandachely k-Signed total dominating family (function) on G. Particularly, a Smarandachely 1-Signed total dominating function or family is called signed total dominating function or family on G. The maximum number of functions in a signed total dominating family on G is the signed total domatic number of G. In this paper, some properties related signed total domatic number and signed total domination number of a graph are studied and found the sign total domatic number of certain class of graphs such as fans, wheels and generalized Petersen graph. Various numerical invariants of graphs concerning domination were introduced by means of dominating functions and their variants (1) and (4). We considered flnite, undirected, simple graphs G = (V;E) with vertex set V (G) and edge set E(G). The order of G is given by n = jV (G)j. If v 2 V (G), then the open neighborhood of v is N(v) = fu 2 V (G)juv 2 E(G)g and the closed neighborhood of v is N(v) = fvg ( N(v). The number dG(v) = d(v) = jN(v)j is the degree of the vertex v 2 V (G), and -(G) is the minimum degree of G. The complete graph and the cycle of order n are denoted by Kn and Cn respectively. A fan and a wheel is a graph obtained from a path and a cycle by adding a new vertex and edges joining it to all the vertices of the path and cycle respectively. The generalized Petersen graph P(n;k) is deflned to be a graph on 2n vertices with V (P(n;k)) = fviui : 1 • ing and E(P(n;k)) =