# On Resolvability- and Domination-Related Parameters of Complete Multipartite Graphs
## 4 Domination-Related Parameters
### Domination Number  
The main purpose of this subsection is to calculate the domination number of complete multipartite graphs. The following theorem provides the exact value of the domination number.  
Figure 5: The complete multipartite graph \(K_{3,3,5}\) with a mixed metric generator (black vertices).  
**Theorem 6**: _Let \(K_{t_{1},t_{2},\ldots,t_{r}}\) be the complete \(r\)-partite graph with \(1\leq t_{1}\leq t_{2}\leq\ldots\leq t_{r}\) and \(\sum\limits_{i=1}^{r}t_{i}=n\). Then_  
\[\gamma(K_{t_{1},t_{2},\ldots,t_{r}})=\left\{\begin{array}{ll}1,&\text{if }t_{1}=1;\\ 2,&\text{Otherwise.}\end{array}\right.\]  
Proof.: Let \(T_{i}\) be the \(i^{\text{th}}\) partite set having size \(t_{i}\), i.e., \(|T_{i}|:=t_{i}\).  
**Case 1:** Assume that \(t_{1}=1\).  
This implies that \(T_{1}\) has only one vertex, say, \(x\). Then, \(D=\{x\}\) is a dominating set, as \(x\) is adjacent to all the vertices in \(T_{i}\) (\(2\leq i\leq r\)) i.e., \(|D\bigcap N(x)|\geq 1\). This implies that \(\gamma(K_{1,t_{2},\ldots,t_{r}})\leq 1\). By definition, for any connected graph \(\Gamma\), \(\gamma(\Gamma)\geq 1\). This shows that \(\gamma(K_{1,t_{2},\ldots,t_{r}})=1\).  
**Case 2:** Assume that \(2\leq t_{1}\leq t_{2}\leq\ldots\leq t_{r}\).  
This implies that every partite set has at least two vertices. Choose a vertex \(x\) from any partite set. Without loss of generality, we may take \(x\in T_{1}\), such that \(D=\{x\}\). Since \(|T_{1}|\geq 2\), it contains at least one other vertex, say \(y\in T_{1}\). Then \(|D\bigcap N(y)|=0\) as \(y\
sim x\) in \(T_{1}\). This implies that \(D\) is not a dominating set of \(K_{t_{1},t_{2},\ldots,t_{r}}\). Since \(T_{1}\) was an arbitrary choice, we obtain that \(\gamma(K_{t_{1},t_{2},\ldots,t_{r}})\geq 2\).  
On the other hand, let us take two elements from different partite sets in \(K_{t_{1},t_{2},\ldots,t_{r}}\) in \(D\). Without loss of generality, we assume that \(D=\{x,y\}\) where \(x\in T_{1}\) and \(y\in T_{2}\). Then, all the vertices in \(T_{i}\) (\(3\leq i\leq r\)) are adjacent to both \(x\) and \(y\), i.e., \(|D\bigcap N(u)|\geq 1\) for any \(u\in T_{i}\) (\(3\leq i\leq r\)). Moreover, open neighborhoods of vertices other then \(x\) and \(y\) in \(T_{1}\) and \(T_{2}\), respectively, also have common elements with \(D\) which are either \(x\) or \(y\). This implies that \(|D\bigcap N(x^{\prime})|=1\) for arbitrary \(x^{\prime}\in T_{1}\) such that \(x\
eq x^{\prime}\) and \(|D\bigcap N(y^{\prime})|=1\) for any \(y^{\prime}\in T_{2}\) such that \(y\
eq y^{\prime}\). This shows that \(D\) is a dominating set of \(K_{t_{1},t_{2},\ldots,t_{r}}\) and \(\gamma(K_{t_{1},t_{2},\ldots,t_{r}})\leq 2\). Combining it with other inequality, we obtain \(\gamma(K_{t_{1},t_{2},\ldots,t_{r}})=2\). This completes the proof.  
The following example illustrates an application of Theorem 6.  
**Example 5**: _Let \(K_{2,3,5}\) be the complete multipartite graph as depicted in Figure 6. Then, its two red vertices form a minimum dominating set. By Theorem 6, we obtain that \(\gamma(K_{3,3,5})=2\)._  
The next subsection studies the locating-dominating number of graphs.