<Li> The metric dimension of an n - vertex graph is n − 1 if and only if it is a complete graph . </Li> <Li> The metric dimension of an n - vertex graph is n − 2 if and only if the graph is a complete bipartite graph K, a split graph K s + K t _̄ (s ≥ 1, t ≥ 2) (\ displaystyle K_ (s) + (\ overline (K_ (t))) (s \ geq 1, t \ geq 2)), or K s + (K 1 ∪ K t) (s, t ≥ 1) (\ displaystyle K_ (s) + (K_ (1) \ cup K_ (t)) (s, t \ geq 1)). </Li> <P> Khuller, Raghavachari & Rosenfeld (1996) prove the inequality n ≤ D β + β (\ displaystyle n \ leq D ^ (\ beta) + \ beta) for any n - vertex graph with diameter D and metric dimension β . This bounds follows from the fact that each vertex that is not in the resolving set is uniquely determined by a distance vector of length β with each entry being an integer between 1 and D (there are precisely D β (\ displaystyle D ^ (\ beta)) such vectors). However, the bound is only achieved for D ≤ 3 (\ displaystyle D \ leq 3) or β = 1 (\ displaystyle \ beta = 1); the more precise bound n ≤ (⌊ 2 D / 3 ⌋ + 1) β + β ∑ i = 1 ⌈ D / 3 ⌉ (2 i − 1) β − 1 (\ displaystyle n \ leq \ left (\ lfloor 2D / 3 \ rfloor + 1 \ right) ^ (\ beta) + \ beta \ sum _ (i = 1) ^ (\ lceil D / 3 \ rceil) (2i - 1) ^ (\ beta - 1)) is proved by Hernando et al. (2010). </P> <P> For specific graph classes, smaller bounds can hold . For example, (Foucaud et al. 2017a) proved bounds of the form n = O (D β 2) (\ displaystyle n = O (D \ beta ^ (2))) for interval graphs and permutation graphs, and bounds of the form n = O (D β) (\ displaystyle n = O (D \ beta)) for unit interval graphs, bipartite permutation graphs and cographs . </P>

The distance between vertices of a connected graph is a metric