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Keywords:
signed total dominating function; signed total domination number; regular graph; circuit; complete graph; complete bipartite graph; Cartesian product of graphs
signed total dominating function; signed total domination number; regular graph; circuit; complete graph; complete bipartite graph; Cartesian product of graphs
Summary:
The signed total domination number of a graph is a certain variant of the domination number. If $v$ is a vertex of a graph $G$, then $N(v)$ is its oper neighbourhood, i.e. the set of all vertices adjacent to $v$ in $G$. A mapping $f: V(G) \rightarrow \lbrace -1, 1\rbrace $, where $V(G)$ is the vertex set of $G$, is called a signed total dominating function (STDF) on $G$, if $\sum _{x \in N(v)} f(x) \ge 1$ for each $v \in V(G)$. The minimum of values $\sum _{x \in V(G)} f(x)$, taken over all STDF’s of $G$, is called the signed total domination number of $G$ and denoted by $\gamma _{\mathrm st}(G)$. A theorem stating lower bounds for $\gamma _{\mathrm st}(G)$ 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 $\gamma _{\mathrm st}(G)$ is not bounded from below in general.
References:
[1] J. E.Dunbar, S. T. Hedetniemi, M. A. Henning and P. J. Slater: Signed domination in graphs. Graph Theory, Combinatorics and Application, Proceedings 7th Internat. Conf. Combinatorics, Graph Theory, Applications, vol. 1, Y. Alavi, A. J. Schwenk (eds.), John Willey & Sons, Inc., 1995, pp. 311–322. MR 1405819
[2] T. W. Haynes, S. T. Hedetniemi and P. J. Slater: Fundamentals of Domination in Graphs. Marcel Dekker Inc., New York-Basel-Hong Kong, 1998. MR 1605684
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