# On Resolvability- and Domination-Related Parameters of Complete Multipartite Graphs
## 2 Preliminaries  
A graph is an ordered pair \(\Gamma(V_{\Gamma},E_{\Gamma})\), where \(V_{\Gamma}\) denotes the point/vertex set and \(E_{\Gamma}\) the line/edge set. Let the non-adjacency (resp. adjacency) of a pair of vertices \(y,z\in V_{\Gamma}\) be denoted by \(y\approx z\) (resp. \(y\sim z\)), implying that \(yz\
otin E_{\Gamma}\) (resp. \(yz\in E_{\Gamma}\)). A graph \(\Gamma(V_{\Gamma},E_{\Gamma})\) is said to be finite if \(V_{\Gamma}\) is a finite set. An edge is called a loop if its end-vertices are the same. Two edges are said to be multiedges if they have the same end-vertices. For \(y\in V_{\Gamma}\), the open (resp. close) neighborhood of \(y\) is \(\mathcal{N}(y)=\{z\in V_{\Gamma}\mid yz\in E_{\Gamma}\}\) (resp. \(\mathcal{N}[y]=\{y\}\cup\{z\in V_{\Gamma}\mid yz\in E_{\Gamma}\}\)). The number \(|\ \mathcal{N}(y)\mid:=\deg_{\Gamma}(y)\) is known to be the degree of \(y\). Contextual understanding of \(\Gamma\) in the text tends to omit \(\Gamma\) from notations \(V_{\Gamma}\), \(\deg_{\Gamma}(y)\) and \(E_{\Gamma}\). A graph is said to be connected if every pair of vertices are connected by a path. It is called undirected if none of its edges has some orientation. Therefore, the binary relation on vertices 'being adjacent' in an undirected graph is symmetric. A graph is said to be simple if it has no loops or multiedges. A complete graph \(K_{n}\) is an \(n\)-vertex graph whose vertices are pairwise adjacent. An independent set \(I\subset V_{\Gamma}\) in a graph \(\Gamma\), is a set of mutually disjoint non-adjacent vertices in \(\Gamma\). A \(r\)-partite graph is a graph whose vertices are or can be partitioned into \(r\) different independent sets. A complete \(r\)-partite graph is an \(r\)-partite graph in which there is an edge between every pair of vertices from different independent sets. A complete \(r\)-partite graph is usually denoted by \(K_{t_{1},t_{2},\ldots,t_{r}}\). Let \(T_{r}\) be the \(r^{\text{th}}\) partite set of \(K_{t_{1},t_{2},\ldots,t_{r}}\) such that \(t_{i}:=\mid T_{i}\mid(1\leq i\leq r)\). Thus, alternatively, a complete \(r\)-partite graph is an \(r\)-partite graph such that \(\forall x\in T_{i},\exists j\
eq i,\exists y\in T_{j},xy\in E_{\Gamma}\) is satisfied. Note that if \(t_{i}=1\) for \(1\leq i\leq r\), then \(K_{t_{1},t_{2},\ldots,t_{r}}=K_{n}\) such that \(\sum\limits_{i=1}^{r}t_{i}=n\). Thus, if we consider non-complete graphs, then we may assume that \(1\leq t_{1}\leq t_{2}\leq\ldots\leq t_{r}\) such that \(t_{1}\geq 1\) and \(t_{i}\geq 2\) for \(2\leq i\leq r\). For more notations and terminologies, we refer the book on molecular topology by [PERSON] et al. [54]. Note that simple graphs have important connection with mathematical chemistry, as described by [PERSON] and [PERSON] [55]. More rich literature on mathematical chemistry and chemical graph theory can be found in [56; 57]. The length of the shortest path between a pair \(y,z\in V_{\Gamma}\) is said to be the distance \(d(y,z)\) between them. Let \(Y=\{y_{k}\}_{k=1}^{\ell}\subseteq V(\Gamma)\) and \(x\in V(\Gamma)\). The distance representation \(r_{Y}(x)\) is the vector \(\big{(}d(x,y_{1})\ldots,d(x,y_{\ell})\big{)}\) of distances from \(x\) to \(y_{k}\)\((1\leq k\leq\ell)\). If all vertices of \(\Gamma\) have distinct distance representations corresponding to \(Y\), the set \(Y\) is said to be a resolving set. The metric dimension \(\beta(\Gamma)\) (or simply \(\operatorname{md}\)) is the smallest cardinality of such a resolving set in \(\Gamma\)[1]. A resolving set of cardinality \(\beta(\Gamma)\) in \(\Gamma\) is said to be a metric generator of \(\Gamma\).