How to uniquely determine your location in a graph? A metric dimension problem

The metric dimension problem was first introduced in 1975 by Slater [35], and independently by Harary and Melter [16] in 1976; however the problem for hypercube was studied (and solved asymptotically) much earlier in 1963 by Erdős and Rényi [10]. A set of vertices S resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. Garey and Johnson [13], and also Khuller et al. [23], showed that determining the metric dimension of an arbitrary graph is an NP-complete problem. The problem is still NP-complete even if we consider some specific families of graphs, such as planar graphs [9] or Gabriel unit disk graphs [18]. Thus research in this area are then constrained towards: characterizing graphs with particular metric dimensions, determining metric dimensions of particular graphs, and constructing algorithm that best approximate metric dimensions. Until today, only graphs of order n with metric dimension 1, n− 3, n− 2, and n− 1 have been characterized [6, 21]. On the other hand, researchers have determined metric dimensions for many particular classes of graphs, such as trees [6, 16, 23], cycles [6], complete multipartite graphs [6, 31], grids [27], wheels [4, 5, 33], fans [5], unicyclic graphs[29], honeycombs [26], circulant graphs [30, 19], Jahangir graphs [36], and Sierṕıski graphs [22]. Recently in 2011, Bailey and Cameron [2] established relationship between the base size of automorphism group of a graph and its metric dimension. This result then motivated researchers to study metric dimensions of distance regular graphs, such as Grassman [3, 15], Johnson, Kneser [1], and bilinear form graphs [12, 14]. There are also some results of metric dimensions of graphs resulting from graph operations, for instance: Cartesian product graphs [27, 23, 5], joint product graphs [4, 5, 33], corona product graphs [37, 20], lexicographic product graphs [32], and amalgamation product graphs [34]. In the area of constructing algorithm that best approximate metric dimensions, recently researchers have utilized integer programming [8], genetic algorithm [24], variable neighborhood search based heuristic [28], and greedy constant factor approximation algorithm [17]. An natural analogue for oriented graphs was introduced by Chartrand, Raines, and Zhang much later in 2000 [7]. Since a directed path from one vertex to another needs