Signed Total Domination Nnumber of a Graph

Abstract

The signed total domination number of a graph is a certain variant of the domination number. If υ is a vertex of a graph G, then N(υ) is its oper neighbourhood, i.e. the set of all vertices adjacent to υ in G. A mapping f: V(G)→-1, 1, where V(G) is the vertex set of G, is called a signed total dominating function (STDF) on G, if $\sum\limits_{x \in N(\upsilon )}^{} {} f(x) \geqslant 1$ for each $\upsilon \in$ V(G). The minimum of values $\sum\limits_{x \in V(G)} {} f(x)$, taken over all STDF's of G, is called the signed total domination number of G and denoted by γ_st(G). A theorem stating lower bounds for γ_st(G) is stated for the case of regular graphs. The values of this number are found for complete graphs, circuits, complete bipartite graphs and graphs on n-side prisms. At the end it is proved that γ_st(G) is not bounded from below in general.