Signed Total Domination Nnumber of a Graph
Bohdan Zelinka · Czechoslovak Mathematical Journal · 2001
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 γst(G). A theorem stating lower bounds for γst(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 γst(G) is not bounded from below in general.