Signed total domination number of a graph

Zelinka, Bohdan

Czechoslovak Mathematical Journal, Tome 51 (2001) no. 2, pp. 225-229.

Résumé

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.

Détail

Classification : 05C35, 05C69
Keywords: signed total dominating function; signed total domination number; regular graph; circuit; complete graph; complete bipartite graph; Cartesian product of graphs