# On Resolvability- and Domination-Related Parameters of Complete Multipartite Graphs
## 2 Preliminaries
In order to explain the concept of a resolving set, let us take the example of the well-known Petersen graph \(P\), see Figure 1. The set \(S=\{v_{1},v_{2},u_{3},u_{4}\}\) is a resolving set of \(P\), since all distance representations of vertices in \(V(P)\setminus S\), i.e.,  
\[r_{S}(u_{1}) =(1,2,2,2); r_{S}(u_{2}) =(2,1,1,2);\] \[r_{S}(u_{5}) =(2,2,2,1); r_{S}(v_{3}) =(1,2,1,2);\] \[r_{S}(v_{4}) =(1,1,2,1); r_{S}(v_{5}) =(2,1,2,2),\]  
are all unique. However, if you observe the distance representations, then we can see that removing the last component from each vector would also keep the representations unique. This implies that the set \(R=S\setminus\{u_{4}\}\) is also a resolving set. It can be shown that there exists no resolving set of cardinality \(2\) in \(P\). Thus, \(\beta(P)=3\). Let \(Y\) be a resolving set. If \(Y^{\prime}=Y\setminus\{y\}\) is also a resolving set for any \(y\in Y\), then \(Y\) is called a fault-tolerant resolving set of \(\Gamma\). Symbolized as \(\beta^{\prime}(\Gamma)\), the fault-tolerant metric dimension (or ftmd) [31] of \(\Gamma\) is the smallest cardinality of such kind of resolving set of \(\Gamma\). A fault-tolerant resolving set of cardinality \(\beta^{\prime}(\Gamma)\) in \(\Gamma\) is said to be a fault-tolerant metric generator of \(\Gamma\). See more on distances in graphs in a book on this title by [PERSON] and [PERSON] [3]. The tree \(T\) in Figure 2 has \(\beta(T)=10\) and \(\beta^{\prime}(T)=14\). The set \(P=(1,2,3,4,\ldots,10)\) (resp. \(Q=P\cup\{y,v,r,s\}\)) forms the metric generator (resp. fault-tolerant metric generator) of \(T\). For \(y\in V_{\Gamma}\) and \(f=ab\in E_{\Gamma}\), the vertex-edge metric/distance is determined as \(d(y,f):=\min\{d(a,y),d(b,y)\}\). A vertex \(y\in V_{\Gamma}\) is known to resolve two edges \(f,g\in E(\Gamma)\) if we have \(d(y,f)\
eq d(y,g)\). An edge-resolving set is a set \(Y=\{y_{k}\}_{k=1}^{\ell}\subseteq V_{\Gamma}\) if all pairwise edges are resolved by some \(y_{k}\in Y\) (\(1\leq k\leq\ell\)). Symbolized as \(\beta_{e}(\Gamma)\), the edge metric  
Figure 1: The Petersen graph \(P\). dimension (or emd) [22] is the smallest cardinality of such a resolving set of \(\Gamma\). An edge resolving set of cardinality \(\beta_{e}(\Gamma)\) in \(\Gamma\) is said to be an edge metric generator of \(\Gamma\). On the other hand, a vertex \(w\in V_{\Gamma}\) is known to resolve \(y,z\in V\cup E\) if \(d(w,y)\
eq d(w,z)\) is satisfied. A mixed-resolving set is a subset \(Y=\{y_{k}\}_{k=1}^{\ell}\subseteq V(\Gamma)\) for which a pair of two elements of \(V\cup E\) is resolved by some vertex of \(Y\). Minimum cardinality \(\beta_{m}(\Gamma)\) of such a mixed-resolving set is known as the mixed-metric dimension (or mmd) [23] of \(\Gamma\). A mixed resolving set of cardinality \(\beta_{m}(\Gamma)\) in \(\Gamma\) is said to be a mixed metric generator of \(\Gamma\). From the definition of an edge resolving set, we obtain that:  
\[\beta_{m}(\Gamma)\geq\max\{\beta(\Gamma),\beta_{e}(\Gamma)\}. \tag{1}\]  
A vertex \(z\in\mathcal{N}(y)\) for an arbitrary \(y\in V_{\Gamma}\) is known as a maximal neighbor of \(y\), if \(\mathcal{N}[y]\subseteq\mathcal{N}[z]\). [PERSON] et al. [23] proved a result as follows:  
Theorem 1: [23] _For an \(n\)-vertex graph \(\Gamma\), we have \(\beta_{m}(\Gamma)=n\) if and only if every vertex of \(\Gamma\) has a maximal neighbor._  
A set \(L\subset V_{\Gamma}\) satisfying \(L\cap\mathcal{N}(y)\
eq\emptyset\) for every \(y\in V_{\Gamma}\setminus L\) is known to dominate (i.e., a dominating set of) \(\Gamma\). The smallest cardinality \(\gamma(\Gamma)\) of such a dominating set is known to be the domination number of \(\Gamma\). A dominating generator is simply a dominating set having cardinality \(\gamma(\Gamma)\). For \(T\subset V_{\Gamma}\), a vertex \(y\in V_{\Gamma}\) is known to dominate \(T\) if \(T\subseteq\mathcal{N}(y)\). If, in addition to being a dominating set, a set \(L\) has distinct distance representation respective to vertices in \(L\), it is known as a metric-locating-dominating (mld) set.