# On Resolvability- and Domination-Related Parameters of Complete Multipartite Graphs
## 3 Resolvability-Related Parameters
### Mixed Metric Dimension  
This section provides a formula for the mmd of complete multipartite graphs. It generalizes a result of [PERSON] et al. [23], who computed the mmd of the complete bipartite case.  
The following result calculates the mmd of the complete \(r\)-partite graph.  
**Theorem 5**.: _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:_  
\[\beta_{m}(K_{t_{1},t_{2},\ldots,t_{r}})=\left\{\begin{array}{ll}\sum\limits _{i=1}^{r}t_{i}-1,&\text{if $t_{i}=2$ for some $i$;}\\ \sum\limits_{i=1}^{r}t_{i}-r,&\text{Otherwise.}\end{array}\right.\]Proof.: By Theorems 2 and 4 and Equation (1), We obtain that \(\beta_{m}(K_{t_{1},t_{2},\ldots,t_{r}})\geq\sum\limits_{i=1}^{r}t_{i}-r\). Let \(T_{i}\) (\(1\leq i\leq r\)) be the \(i^{\text{th}}\) partite set with \(|T_{i}|=t_{i}\). First, we assume that \(t_{1}=2\). Suppose \(\beta_{c}(K_{2=t_{1},t_{2},\ldots,t_{r}})=\sum\limits_{i=1}^{r}t_{i}-1\) and assume \(L\) is the mixed resolving set of the smallest cardinality of \(K_{2,t_{2},\ldots,t_{r}}\). Notice that any edge resolving or simply resolving sets have to comprise \(t_{i}-1\) elements of \(T_{i}\), where \(1\leq i\leq r\). We deduce that \(\mid T_{1}\cap L\mid=1\) and \(|T_{i}\cap L|=t_{i}-1\) for \(i\in\{2,3,\ldots,r\}\). Let \(y\in T_{1}\cap L\) and \(z\in T_{i}\cap L\) for \(i\in\{2,3,\ldots,r\}\). Then, \(d(y,y)=0\) and \(d(y,z)=1\). In addition, since \(g=yz\in E_{K_{t_{1},t_{2},\ldots,t}}\), we have the vertex-edge distance \(d(g,y)=0\) and \(d(g,z)=0\). This shows that both \(y\) and \(g\) are not distinguished by \(L\), which then leads to a contradiction to the minimality of \(L\). A similar argument holds if any of the \(t_{i}=2\) for \(2\leq i\leq r\). This shows first part of the result by using Theorem 1.  
Next, we assume that \(t_{i}\geq 3\) for \(1\leq i\leq r\). Assume \(L\) to be a minimum set comprising the \(\sum\limits_{i=1}^{r}t_{i}-r\) element in a way it considers all elements of \(K_{t_{1},t_{2},\ldots,t_{r}}\) except one from every partite set. Note that \(L\) must distinguish all vertices/edges pairs of \(K_{t_{1},t_{2},\ldots,t_{r}}\), since, by assumption, \(L\) is a minimum resolving and edge-resolving set. However, as \(d(g,y)=\{0,1\}\) for any \(g\in E_{K_{t_{1},t_{2},\ldots,t_{r}}}\) and \(y\in K_{t_{1},t_{2},\ldots,t_{r}}\), and there exists at least one element in \(L\) which is at distance \(2\) from other vertices of \(K_{t_{1},t_{2},\ldots,t_{r}}\), since \(t_{i}\geq 3\) for \(1\leq i\leq r\), it is convenient to see that \(L\) distinguishes all vertex-edge pair of \(K_{t_{1},t_{2},\ldots,t_{r}}\). This implies the minimality of \(L\) with cardinality \(\sum\limits_{i=1}^{r}t_{i}-r\) and this completes the result.  
The following example illustrates an application of Theorem 5.  
**Example 4**.: _Let \(K_{3,3,5}\) be the complete multipartite graph as depicted in Figure 5. Then, its eight black vertices form a mixed metric generator. By Theorem 5, we obtain that \(\beta(K_{3,3,5})=11-3=8\)._  
It is important to remark the following:  
**Remark 4**.: _It is worth noting that Theorem 5 generalizes Proposition 4.2 [23] by [PERSON] et al. [23]._