# On Resolvability- and Domination-Related Parameters of Complete Multipartite Graphs
## 3 Resolvability-Related Parameters
### Fault-Tolerant Metric Dimension  
In Theorem 7 [53], [PERSON] et al. characterized \(n\)-vertex graphs with \(\beta^{\prime}(\Gamma)=n\). In particular, they showed the complete graphs are the only graphs satisfying \(\beta^{\prime}(\Gamma)=n\). They used the classification of \(n\)-ordered graphs with \(\beta(\Gamma)=n-1\), together with  
\[\beta^{\prime}(\Gamma)\geq\beta(\Gamma)+1, \tag{4}\]  
to show Theorem 7 [53]. However, relationship in \(\beta(\Gamma)\leq n-1\) does not ensure a relationship in (4). This fact makes the proof of Theorem 7 [53] invalid.  
The next theorem proves that the complete multipartite graphs has \(\beta^{\prime}(\Gamma)=n\), and thus, providing an infinite family of counterexamples to Theorem 7 [53].  
**Theorem 3**.: _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:_  
\[\beta^{\prime}(K_{t_{1},t_{2},\ldots,t_{r}})=\sum\limits_{i=1}^{r}t_{i}=n.\]  
Figure 3: The complete multipartite graph \(K_{2,3,5}\) with a metric generator (black vertices).  
Proof.: By definition of the fault-tolerant metric dimension, we know that \(2\leq\beta^{\prime}(\Gamma)\leq n\) for any connected graph \(\Gamma\) on \(n\) vertices. Thus, we only need to show that:  
\[\beta^{\prime}(K_{t_{1},t_{2},\ldots,t_{r}})\geq n=\sum_{i=1}^{r}t_{i}.\]  
On the contrary, we assume that this is not true. This implies that there exists a fault-tolerant resolving set, say \(F\), on \(n-1\) vertices. Then, let us assume that we have \(V(K_{t_{1},t_{2},\ldots,t_{r}})\setminus F=\{x\}\). By definition of a fault-tolerant resolving set, the deletion of an arbitrary vertex in \(F\) leaves a set which is also a resolving set. Without loss of generality, we assume that \(x\in T\), i.e., an arbitrary partite set. Then, we delete any other vertex from \(T\), say \(y\), such that \(x\
eq y\) and let \(F^{\prime}=F\setminus\{y\}=V(K_{t_{1},t_{2},\ldots,t_{r}})\setminus\{x,y\}\). Note that \(r_{F^{\prime}}(x)=r_{F^{\prime}}(y)\). This shows that \(F^{\prime}\) is not a resolving set. Then, it implies that \(F\) is not a fault-tolerant resolving set, which causes a contradiction. Thus, there exists no fault-tolerant resolving set of cardinality \(n-1\). This completes the proof.  
The following example illustrates an application of Theorem 3.  
**Example 2**.: _Let \(K_{2,3,5}\) be the complete multipartite graph as given in Figure 4. Then, by Theorem 3, all of its vertices form a fault-tolerant metric generator and we obtain that \(\beta(K_{2,3,5})=10\)._  
The following remark provides additional importance of Theorem 3.  
**Remark 2**.: _Note that Proposition 3 provides a counter example to Theorem 7[53]. This shows that the characterization in Theorem 7[53] is incorrect._  
Based on this remark, we raise the following open problem.  
**Problem 1**.: _Let \(\Gamma\) be a connected graph on \(n\) vertices such that:_  
\[\beta^{\prime}(\Gamma)=n. \tag{5}\]  
_Are the complete graphs and the complete multipartite graphs the only graphs for which (5) holds? If yes, then characterize this extremal case, and if no, then provide additional examples satisfying (5)._