# Domination and Independent Domination in Extended Supergrid Graphs +
## 4 The Domination and Independent Domination Numbers of Rectangular Triangular-Supergrid Graphs
In any case, we construct a (independent) dominating set \(D\) of \(T_{m\times n}\) with size \(\lceil\frac{m}{3}\rceil(\lfloor\frac{n}{2}\rfloor+1)=\lceil\frac{m}{3}\rceil \lceil\frac{n+1}{2}\rceil\) or less. For example, Figure 13b shows the constructed (independent) dominating set \(D\) of \(T_{8\times 13}\) with size \(2\times 7+6=(\lceil\frac{m}{3}\rceil-1)\times\lceil\frac{13+1}{2}\rceil+ \lceil\frac{2\times 13}{5}\rceil\leqslant\lceil\frac{8}{3}\rceil(\lceil\frac{3+1}{2}\rceil\). That is, we obtain an upper bound \(\lceil\frac{m}{3}\rceil\lceil\frac{n+1}{2}\rceil\) of \(\gamma(T_{m\times n})\) (\(\gamma_{\text{ind}}(T_{m\times n})\)), and hence the lemma holds true. Figure 13: **(a)** Partitioning of \(T_{m\times n}\) into \(\lceil\frac{m}{3}\rceil\) disjoint subgraphs, and **(b)** an (independent) dominating set of \(T_{m\times n}\). Solid dashed lines indicate separations and filled circles indicate vertices in the (independent) dominating set. By the above lemma, we can see that each vertex in the constructed (independent) dominating set of \(T_{m\times n}\) almost dominates six vertices. By the structure of \(T_{m\times n}\), each vertex dominates at most seven vertices, including its six neighbors and itself. However, there exists no dominating set of \(T_{m\times n}\) in which its each vertex dominates seven vertices. Thus, \(\lceil\frac{nm}{7}\rceil<\gamma(T_{m\times n})\leqslant\gamma_{\text{ind}}(T_{m \times n})\). Because an independent dominating set of a graph is a dominating set of the graph, \(\gamma(T_{m\times n})\) provides a trivial lower bound of \(\gamma_{\text{ind}}(T_{m\times n})\). Then, \(\lceil\frac{nm}{7}\rceil<\gamma(T_{m\times n})\leqslant\gamma_{\text{ind}}(T_ {m\times n})\leqslant\lceil\frac{n}{3}\rceil\lceil\frac{n+1}{2}\rceil\) for \(n\geqslant m\geqslant 4\). Thus, we obtain a tight upper bound of \(\gamma(T_{m\times n})\) and \(\gamma_{\text{ind}}(T_{m\times n})\). Using Lemmas 9-11, we can conclude the following theorem. Theorem 4.1: _Let \(T_{m\times n}\) be a rectangular triangular-supergrid graph with \(n\geqslant m\geqslant 2\). Then, \(\lceil\frac{nm}{7}\rceil<\gamma(T_{m\times n})\leqslant\gamma_{\text{ind}}(T_ {m\times n})\) if \(n\geqslant m\geqslant 4\), and_  
\[\gamma(T_{m\times n})(\gamma_{\text{ind}}(T_{m\times n}))\left\{ \begin{array}[ ]{ll}=\lceil\frac{2n}{3}\rceil&\text{, if }m=2;\\ =\lceil\frac{n+1}{2}\rceil&\text{, if }m=3;\\ \leqslant\lceil\frac{n}{3}\rceil\lceil\frac{n+1}{2}\rceil&\text{, otherwise.}\end{array} \right.\]  
Proof. : By Lemma 9, \(\gamma(T_{2\times n})=\gamma_{\text{ind}}(T_{2\times n})=\lceil\frac{2n}{3}\rceil\). By Lemma 10, \(\gamma(T_{3\times n})=\gamma_{\text{ind}}(T_{3\times n})=\lceil\frac{n+1}{2}\rceil\). Lemma 11 provides that \(\gamma(T_{m\times n})\leqslant\lceil\frac{m}{3}\rceil\lceil\frac{n+1}{2}\rceil\) and \(\gamma_{\text{ind}}(T_{m\times n})\leqslant\lceil\frac{m}{3}\rceil\lceil \frac{n+1}{2}\rceil\) for \(n\geqslant m\geqslant 4\). In addition, \(\lceil\frac{nm}{7}\rceil<\gamma(T_{m\times n})\leqslant\gamma_{\text{ind}}(T_ {m\times n})\) if \(n\geqslant m\geqslant 4\). Thus, the theorem holds true.