# On Resolvability- and Domination-Related Parameters of Complete Multipartite Graphs
## 1 Introduction  
This paper takes only finite, connected and simple graphs into account. The reader is referred to Section 2 for undefined notations and terminologies. The concept of metric dimension was first defined independently by [PERSON] in 1975 [PERSON] (1975 [PERSON] (1975)), and later, it was discussed in 1976 by [PERSON] and [PERSON] [PERSON] and [PERSON] (1976). It has been studied widely since then. Metric dimension has diverse applications in fields such as graph theory [PERSON] and [PERSON] (1977), robot navigation [PERSON] and [PERSON] (1977), chemistry [PERSON] and [PERSON] (1977), telecommunication networks [PERSON] and [PERSON] (1977), combinatorial optimization [PERSON] and [PERSON] (1977), geographical routing protocols [PERSON] and [PERSON] (1977), sociology [PERSON] and [PERSON] (1977) and many more. The metric dimension has been studied extensively since it has been introduced. For instance, in [PERSON] and [PERSON] (1977); [PERSON] and [PERSON] (1977); [PERSON] and [PERSON] (1977); [PERSON] and [PERSON] (1977); [PERSON] and [PERSON] (1977); [PERSON] and [PERSON] (1977), the authors studied the metric dimension of certain infinite families of graphs. Other variants of metric dimension such as the edge metric and the mixed-metric dimension of graphs have been defined by [PERSON] et al. [PERSON] et al. (2015) and [PERSON] et al. [PERSON] et al. (2015), respectively. The edge metric dimension for various families of graphs has been studied in [PERSON] et al. (2015); [PERSON] et al. (2015); [PERSON] et al. (2015); [PERSON] et al. (2015), among others. The mixed metric dimension for various families of graphs has been investigated in [PERSON] et al. (2015); [PERSON] et al. (2015); [PERSON] et al. (2015), among others. The fault-tolerant metric dimension of graphs has been introduced by [PERSON] et al. [PERSON] et al. (2015) back in 2008 as a natural extension of the metric dimension. The reader is referred to [PERSON] et al. (2015); [PERSON] et al. (2015); [PERSON] et al. (2015); [PERSON] et al. (2015) for various mathematical properties of fault-tolerant resolvability in graphs. We refer the interested readers to the book by [PERSON] et al. [PERSON] et al. (2015), which provides, until 1980, a brief overview of the results regarding domination in graphs. [PERSON] et al. [PERSON] et al. (2015) considered trees for their total domination and the binary location-domination numbers. Minimum \(\ell\)-locating-dominating as well as \(\ell\)-identifying sets/representations in chains and cyclic graphs have been constructed by [PERSON] et al. [PERSON] et al. (2015). Sharp upper and lower bounds on the minimality of \(\ell\)-locating-dominating sets for general graph were also found. For further reading on these contemporary domination-related parameters, we refer to [PERSON] et al. (2015); [PERSON] et al. (2015). [PERSON] [41] and [PERSON] et al. [42; 43] introduced the concepts of the open-neighborhood location-domination number and the fault-tolerant location-domination number, respectively, and found results on these parameters for trees. The location-domination number for certain classes of convex polytopes was investigated recently by [PERSON] et al. [8] and [PERSON] et al. [44]. They found its exact values for some families of convex polytopes and tight upper bounds were found for other families. The reader is suggested to read [40; 45; 46; 47] for more details on these domination-related graph-theoretic parameters. Generalized Petersen graphs are of prime importance in graph theory and they provide counterexamples to many graph-theoretic conjectures. They were considered by a number of researchers in [48; 49; 50; 51; 52] for their parameters related to domination in graphs. Taking the extensive literature on the resolvability- and domination-related parameters of graphs into account, in this paper, we find exact values of the metric, edge metric, fault-tolerant metric, and mixed metric dimensions of complete multipartite graphs.