# On Resolvability- and Domination-Related Parameters of Complete Multipartite Graphs
## 3 Resolvability-Related Parameters
### Edge Metric Dimension  
In this subsection, we determine the exact value of the edge metric dimension of the complete multipartite graphs.  
Figure 4: The complete multipartite graph \(K_{2,3,5}\) with the fault-tolerant metric generator (all vertices).  
**Theorem 4**.: _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:_  
\[\beta_{e}(K_{t_{1},t_{2},\ldots,t_{r}})=\sum\limits_{i=1}^{r}t_{i}-r=n-r.\]  
Proof.: Let \(T_{i}\) (\(1\leq i\leq r\)) be the \(i^{\text{th}}\) partite set with \(|T_{i}|=t_{i}\). First, we show that \(\beta_{e}(K_{t_{1},t_{2},\ldots,t_{r}})\geq\sum\limits_{i=1}^{r}t_{i}-r\). In order to show that, first, we need to prove that there exists an edge resolving set of cardinality \(\sum\limits_{i=1}^{r}t_{i}-r\) in \(K_{t_{1},t_{2},\ldots,t_{r}}\). Let \(x_{i}\in T_{i}\) be an arbitrary vertex in \(T_{i}\), where \(2\leq i\leq r\). Assume \(S=V(K_{t_{1},t_{2},\ldots,t_{r}})\setminus\{x_{1},x_{2},\ldots,x_{r}\}\) such that \(\mid S\mid=\sum\limits_{i=1}^{r}t_{i}-r\). We show that \(S\) is an edge resolving set in \(K_{t_{1},t_{2},\ldots,t_{r}}\). Note that we have \(V(K_{t_{1},t_{2},\ldots,t_{r}})\setminus S=\{x_{1},x_{2},\ldots,x_{r}\}\).  
Next, we assume that one of the partite sets contains at least two elements of \(x_{i}\) (\(1\leq i\leq r\)). Without loss of generality, we assume \(Y\) to be that set and \(x_{1},\ x_{2}\in Y\). Let \(A\) be another partite set different from \(Y\). For any \(x\in A\) such that \(x\in S\), consider the edges \(e=xx_{1}\) and \(f=xx_{2}\). Note that both \(e\) and \(f\) have distances \(0\) from \(x\) and one from any other element in \(S\). Thus, both \(e\) and \(f\) have the same edge distance representations and this causes a contradiction to the fact that \(S\) is an edge resolving set. We may proceed with other partite sets in a similar way and it follows that any edge resolving set must contain all but (maybe) one element of every partite set. Since there exactly \(r\) number of partite sets, we obtain that \(\beta_{e}(K_{t_{1},t_{2},\ldots,t_{r}})\geq\sum\limits_{i=1}^{r}t_{i}-r\).  
On the contrary, let \(x_{i}\in T_{i}\) where \(1\leq i\leq r\) and let \(S=V(K_{t_{1},t_{2},\ldots,t_{r}})\setminus\{x_{1},x_{2},\ldots,x_{r}\}\). It can easily be checked that \(S\) is an edge resolving set as any two edges in \(K_{t_{1},t_{2},\ldots,t_{r}}\) have different edge distance representations. Therefore, we obtain that \(\beta_{e}(K_{t_{1},t_{2},\ldots,t_{r}})\leq\sum\limits_{i=1}^{r}t_{i}-r\). The two inequalities complete the proof.  
The following example explains an application of Theorem 4.  
**Example 3**.: _Let \(K_{2,3,5}\) be the complete multipartite graph as given in Figure 3. Then, its seven black vertices form an edge metric generator, i.e., a minimum edge resolving set of \(K_{2,3,5}\). By Theorem 2, we obtain that \(\beta_{e}(K_{2,3,5})=10-3=7\)._  
The following remark gives additional importance of Theorem 4.  
**Remark 3**.: _It is worthy to mention that Theorem 4 generalizes [PERSON] et al. (Remark 2[22]) from the complete bipartite graphs to the complete multipartite graphs._