In graph theory, the metric dimension of a graph G is the minimum cardinality of a subset S of vertices such that all other vertices are uniquely determined by their distances to the vertices in S. Finding the metric dimension of a graph is an NP-hard problem; the decision version, determining whether the metric dimension is less than a given value, is NP-complete.
of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple
The metric dimension of G is the minimum cardinality of a resolving set for G. A resolving set containing a minimum number of vertices is called a basis (or reference set) for G. Resolving sets for graphs were introduced independently by Slater (1975) and Harary & Melter (1976), while the concept of a resolving set and that of metric dimension were defined much earlier in the more general context of metric spaces by Blumenthal in his monograph Theory and Applications of Distance Geometry.
Graphs are special examples of metric spaces with their intrinsic path metric.
If a tree is a path, its metric dimension is one.
Otherwise, let L denote the set of leaves, degree-one vertices in the tree.
Let K be the set of vertices that have degree greater than two, and that are connected by paths of degree-two vertices to one or more leaves.
A basis of this cardinality may be formed by removing from L one of the leaves associated with each vertex in K.[1] The same algorithm is valid for the line graph of the tree, and thus any tree and its line graph have the same metric dimension.
[2] In Chartrand et al. (2000), it is proved that: Khuller, Raghavachari & Rosenfeld (1996) prove the inequality
for any n-vertex graph with diameter
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
For specific graph classes, smaller bounds can hold.
For example, Beaudou et al. (2018) proved that
The authors Foucaud et al. (2017a) proved bounds of the form
Deciding whether the metric dimension of a graph is at most a given integer is NP-complete.
[8] For any fixed constant k, the graphs of metric dimension at most k can be recognized in polynomial time, by testing all possible k-tuples of vertices, but this algorithm is not fixed-parameter tractable (for the natural parameter k, the solution size).
Answering a question posed by Lokshtanov (2010), Hartung & Nichterlein (2013) show that the metric dimension decision problem is complete for the parameterized complexity class W[2], implying that a time bound of the form nO(k) as achieved by this naive algorithm is likely optimal and that a fixed-parameter tractable algorithm (for the parameterization by k) is unlikely to exist.
Deciding whether the metric dimension of a tree is at most a given integer can be done in linear time[10] Other linear-time algorithms exist for cographs,[5] chain graphs,[11] and cactus block graphs[12] (a class including both cactus graphs and block graphs).
The problem may be solved in polynomial time on outerplanar graphs.
[4] It may also be solved in polynomial time for graphs of bounded cyclomatic number,[5] but this algorithm is again not fixed-parameter tractable (for the parameter "cyclomatic number") because the exponent in the polynomial depends on the cyclomatic number.
There exist fixed-parameter tractable algorithms to solve the metric dimension problem for the parameters "vertex cover",[13] "max leaf number",[14] and "modular width".
[9] Graphs with bounded cyclomatic number, vertex cover number or max leaf number all have bounded treewidth, however it is an open problem to determine the complexity of the metric dimension problem even on graphs of treewidth 2, that is, series–parallel graphs.
[9] The metric dimension of an arbitrary n-vertex graph may be approximated in polynomial time to within an approximation ratio of
by expressing it as a set cover problem, a problem of covering all of a given collection of elements by as few sets as possible in a given family of sets.
[15] In the set cover problem formed from a metric dimension problem, the elements to be covered are the
pairs of vertices to be distinguished, and the sets that can cover them are the sets of pairs that can be distinguished by a single chosen vertex.
The approximation bound then follows by applying standard approximation algorithms for set cover.
An alternative greedy algorithm that chooses vertices according to the difference in entropy between the equivalence classes of distance vectors before and after the choice achieves an even better approximation ratio,
cannot be achieved in polynomial time for any
[16] The latter hardness of approximation still holds for instances restricted to subcubic graphs,[13] and even to bipartite subcubic graphs.