# On Resolvability- and Domination-Related Parameters of Complete Multipartite Graphs
## 4 Domination-Related Parameters
### Locating-Dominating Number
Altogether, we obtain that \(S\) must contain exactly \(a+b-2\) vertices, which completes the proof. The next proposition computes the exact value of the locating-dominating number of the complete tripartite graphs. **Proposition 2**. : _Let \(K_{a,b,c}\) be the complete tripartite graph with \(1\leq a\leq b\leq c\). Then, \(\gamma_{l-d}(K_{a,b,c})=a+b+c-3\)._Proof. : Let \((A,B,C)\) be the tripartition of \(V(K_{a,b,c})\), such that \(|A|=a,\ |B|=b\) and \(|C|=c\). For arbitrary vertices \(x\in A,\ y\in B\) and \(z\in C\), let \(S=A\setminus\{x\}\bigcup B\setminus\{y\}\bigcup C\setminus\{z\}\). Note that \(S\) is a locating-dominating set of \(K_{a,b,c}\), since \(S\bigcap N[x]=B\setminus\{y\}\bigcup C\setminus\{z\}\), \(S\bigcap N[y]=A\setminus\{x\}\bigcup C\setminus\{z\}\) and \(S\bigcap N[z]=A\setminus\{x\}\bigcup B\setminus\{y\}\) are non-empty and mutually disjoint. Thus, we obtain that \(\gamma_{l-d}(K_{a,b,c})\leq|S|=a+b+c-3\). In order to complete the proof, we need to show that \(\gamma_{l-d}(K_{a,b,c})\geq a+b+c-3\). We show it by proving that \(\gamma_{l-d}(K_{a,b,c})>a+b+c-4\). On contrary, we assume that \(\gamma_{l-d}(K_{a,b,c})=a+b+c-4\). Thus, there exists a locating-dominating set of cardinality four and let \(S\) be that set. Let \(V(K_{a,b,c})\setminus S=\{w,x,y,z\}\). We distinguish the following possible cases for the vertices \(w,\ x,\ y\), and \(z\) as follows:  
**Case 1:** All \(w,\ x,\ y\), and \(z\) belong to one partite set. Without loss of generality, we may assume that \(\{w,x,y,z\}\subset A\). Assuming it would generate \(S\bigcap N[w]=S\bigcap N[x]=S\bigcap N[y]=S\bigcap N[z]=B\bigcup C\), which causes a contradiction to the fact that \(S\) is a locating-dominating set of \(K_{a,b,c}\). **Case 2:** All \(w,\ x,\ y\), and \(z\) belong to exactly two partite sets. _Subcase 2.1:_ One partite set is singleton. Without loss of generality, we may assume that \(\{w,x,y\}\subset A\) and \(z\in B\). Then, the same intersections \(S\bigcap N[w]=S\bigcap N[x]=S\bigcap N[y]=C\bigcup B\setminus\{z\}\) cause a contradiction that \(S\) is a binary-locating dominating set. _Subcase 2.2:_ Both partite sets have cardinality \(2\). Similar to the previous cases, without loss of generality, we may assume that \(\{w,x\}\subset A\) and \(\{y,z\}\subset B\). In this case, we obtain that \(S\bigcap N[w]=S\bigcap N[x]=C\bigcup B\setminus\{y,z\}\) and \(S\bigcap N[y]=S\bigcap N[z]=C\bigcup A\setminus\{w,x\}\), which is a contradiction again. **Case 3:** All \(w,\ x,\ y\), and \(z\) belong to exactly three partite sets. Without loss of generality, we may assume that \(w\in A\), \(x\in B\), and \(\{y,z\}\in C\). Assuming this would generate \(S\bigcap N[z]=A\setminus\{w\}\bigcup B\setminus\{x\}=S\bigcap N[y]\). Thus, the two intersections are not mutually disjoint which causes a contradiction. Combining all the possible case, we find that \(S\) is not a locating-dominating set of \(K_{a,b,c}\). Since \(S\) was an arbitrary set of cardinality four, we obtain that \(\gamma_{l-d}(K_{a,b,c})>a+b+c-4\geq a+b+c-3\). This completes the proof. Next, we show the main result of this subsection. **Theorem 8**. : _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:_  
\[\gamma_{l-d}(K_{t_{1},t_{2},\ldots,t_{r}})=\sum\limits_{i=1}^{r}t_{i}-r. \tag{6}\]  
Proof. : We prove the result by applying induction on \(r\), which is the number of partite sets in \(K_{t_{1},t_{2},\ldots,t_{r}}\). By Propositions 1 and 2, the result is true for \(r=2\) and \(r=3\), respectively. Now, let the assertion be valid for \(r=k\) as an induction step. We show that the result is true for \(r=k+1\) to complete the proof. Let \(T_{i}\) (\(1\leq i\leq r\)) be the \(i^{\text{th}}\) partite set of \(K_{t_{1},t_{2},\ldots,t_{r}}\), where \(|T_{i}|:=t_{i}\).