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| In [[graph theory]], the '''metric dimension''' of a graph ''G'' is the minimum number of vertices in a subset ''S'' of ''G'' 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]].
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| == Detailed definition ==
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| For an ordered subset <math>W = \{w_1, w_2,\dots w_k\}</math> of vertices and a vertex ''v'' in a connected graph ''G'', the representation of ''v'' with respect to ''W'' is the ordered ''k''-tuple <math>r(v|W) = (d(v,w_1), d(v,w_2),\dots,d(v,w_k))</math>, where ''d''(''x'',''y'') represents the distance between the vertices ''x'' and ''y''. The set ''W'' is a resolving set (or locating set) for ''G'' if every two vertices of ''G'' have distinct representations. 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 were introduced independently by {{harvtxt|Slater|1975}} and {{harvtxt|Harary|Melter|1976}}.
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| ==Trees==
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| {{harvtxt|Slater|1975}} provides the following simple characterization of the metric dimension of a [[tree (graph theory)|tree]]. If the tree is a path, its metric dimension is one. Otherwise, let ''L'' denote the set of degree-one vertices in the tree (usually called leaves, although Slater uses that word differently). 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. Then the metric dimension is |''L''| − |''K''|. A basis of this cardinality may be formed by removing from ''L'' one of the leaves associated with each vertex in ''K''.
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| ==Properties==
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| In {{harvtxt|Chartrand|Eroh|Oellermann|2000}}, it is proved that:
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| * The metric dimension of a graph {{mvar|G}} is 1 if and only if {{mvar|G}} is a path.
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| * The metric dimension of an {{mvar|n}}-vertex graph is {{math|''n'' − 1}} if and only if it is a [[complete graph]].
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| * The metric dimension of an {{mvar|n}}-vertex graph is {{math|''n'' − 2}} if and only if the graph is a [[complete bipartite graph]] {{math|''K''<sub>''s'', ''t''</sub>}}, a [[split graph]] <math>K_s+\overline{K_t} (s\geq 1, t\geq 2)</math>, or <math>K_s+(K_1\cup K_t) (s,t\geq 1) </math>.
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| {{harvtxt|Khuller|Raghavachari|Rosenfeld|1996}} prove the inequality <math> n\leq D^{\beta-1}+\beta</math> for any {{mvar|n}}-vertex graph with [[diameter (graph theory)|diameter]] {{mvar|D}} and metric dimension β.
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| ==Computational complexity==
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| For any 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 [[parameterized complexity|fixed-parameter tractable]]. Answering a question posed by {{harvtxt|Lokshtanov|2010}}, {{harvtxt|Hartung|Nichterlein|2012}} show that metric dimension is complete for the parameterized complexity class W[2], implying that a time bound of the form ''n''<sup>O(''k'')</sup> as achieved by this naive algorithm is likely optimal and that a fixed-parameter tractable algorithm (parameterized by the metric dimension) is unlikely to exist.
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| The metric dimension of an arbitrary ''n''-vertex graph may be approximated in polynomial time to within an [[Approximation algorithm|approximation ratio]] of <math>2\log n</math> 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]] {{harv|Khuller|Raghavachari|Rosenfeld|1996}}. In the set cover problem formed from a metric dimension problem, the elements to be covered are the <math>\tbinom{n}{2}</math> 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, <math>\log n+\log\log_2 n+1</math> {{harv|Hauptmann|Schmied|Viehmann|2012}}. This approximation ratio is close to best possible, as under standard complexity-theoretic assumptions a ratio of <math>(1-\epsilon)\log n</math> cannot be achieved in polynomial time for any <math>\epsilon>0</math> {{harv|Hauptmann|Schmied|Viehmann|2012}}.
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| Metric dimension remains NP-complete for bounded-degree [[planar graph]]s {{harv|Díaz|Pottonen|Serna|van Leeuwen|2012}}. It is also NP-complete for [[split graph]]s, [[bipartite graph]]s and their [[complement (graph theory)|complements]], and [[line graph]]s of bipartite graphs {{harv|Epstein|Levin|Woeginger|2012}}. It may be solved in polynomial time on [[outerplanar graph]]s {{harv|Díaz|Pottonen|Serna|van Leeuwen|2012}} and on [[cograph]]s {{harv|Epstein|Levin|Woeginger|2012}}. It may also be solved in polynomial time for graphs of bounded [[cyclomatic number]], but this algorithm is again not fixed-parameter tractable because the exponent in the polynomial depends on the cyclomatic number {{harv|Epstein|Levin|Woeginger|2012}}.
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| ==References==
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| *{{citation|first1=P.|last1=Buczkowski|first2=G.|last2=Chartrand|author2-link=Gary Theodore Chartrand|first3=C.|last3=Poisson|first4=P.|last4=Zhang|title=On ''k''-dimensional graphs and their bases|journal=Periodica Mathematica Hungarica|volume=46|issue=1|pages=9–15|year=2003|doi=10.1023/A:1025745406160|mr=1975342}}.
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| *{{citation|first1=G.|last1=Chartrand|author1-link=Gary Theodore Chartrand|first2=L.|last2=Eroh|first3=M. A.|last3=Johnson|first4=O. R.|last4=Oellermann|title=Resolvability in graphs and the metric dimension of a graph|journal=Discrete Applied Mathematics|volume=105|issue=1–3|doi=10.1016/S0166-218X(00)00198-0|pages=99–113|year=2000|mr=1780464}}.
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| *{{citation|first1=J.|last1=Díaz|first2=O.|last2=Pottonen|first3=M. J.|last3=Serna|first4=E. J.|last4=van Leeuwen|contribution=On the complexity of metric dimension|title=[[European Symposium on Algorithms|Algorithms – ESA 2012: 20th Annual European Symposium, Ljubljana, Slovenia, September 10-12, 2012, Proceedings]]|volume=7501|series=Lecture Notes in Computer Science|pages=419–430|publisher=Springer|year=2012|url=http://www.lsi.upc.edu/~diaz/papersd/ESA-proc2.pdf|arxiv=1107.2256|editor1-first=Leah|editor1-last=Epstein|editor2-first=Paolo|editor2-last=Ferragina|doi=10.1007/978-3-642-33090-2_37}}.
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| *{{citation|first1=Leah|last1=Epstein|first2=Asaf|last2=Levin|first3=Gerhard J.|last3=Woeginger|contribution=The (weighted) metric dimension of graphs: hard and easy cases|title=Graph-Theoretic Concepts in Computer Science: 38th International Workshop, WG 2012, Jerusalem, Israel, June 26-28, 2012, Revised Selected Papers|series=Lecture Notes in Computer Science|editor1-first=Martin Charles|editor1-last=Golumbic|editor1-link=Martin Charles Golumbic|editor2-first=Michal|editor2-last=Stern|editor3-first=Avivit|editor3-last=Levy|editor4-first=Gila|editor4-last=Morgenstern|volume=7551|year=2012|pages=114–125|doi=10.1007/978-3-642-34611-8_14}}.
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| *{{citation|author1-link=Michael R. Garey|first1=M. R.|last1=Garey|author2-link=David S. Johnson|first2=D. S.|last2=Johnson | year = 1979 | title = [[Computers and Intractability: A Guide to the Theory of NP-Completeness]] | publisher = W.H. Freeman | isbn = 0-7167-1045-5}} A1.5: GT61, p. 204.
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| *{{citation|first1=F.|last1=Harary|author1-link=Frank Harary|first2=R. A.|last2=Melter|title=On the metric dimension of a graph|journal=[[Ars Combinatoria (journal)|Ars Combinatoria]]|volume=2|pages=191–195|year=1976|mr=0457289}}.
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| *{{citation|first1=Sepp|last1=Hartung|first2=André|last2=Nichterlein|title=On the parameterized and approximation hardness of metric dimension|year=2012|arxiv=1211.1636}}.
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| *{{citation
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| | last1 = Hauptmann | first1 = Mathias
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| | last2 = Schmied | first2 = Richard
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| | last3 = Viehmann | first3 = Claus
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| | doi = 10.1016/j.jda.2011.12.010
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| | journal = Journal of Discrete Algorithms
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| | mr = 2922072
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| | pages = 214–222
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| | title = Approximation complexity of metric dimension problem
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| | volume = 14
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| | year = 2012}}.
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| *{{citation|last1=Khuller|first1=S.|last2=Raghavachari|first2=B.|last3=Rosenfeld|first3=A.|title=Landmarks in graphs|journal=Discrete Applied Mathematics|volume=70|issue=3|doi=10.1016/0166-218X(95)00106|pages=217–229|year=1996}}.
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| *{{citation
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| | last = Lokshtanov | first = Daniel
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| | editor1-last = Demaine | editor1-first = Erik D. | editor1-link = Erik Demaine
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| | editor2-last = Hajiaghayi | editor2-first = MohammadTaghi
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| | editor3-last = Marx | editor3-first = Dániel
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| | contribution = Open problems – Parameterized complexity and approximation algorithms: Metric Dimension
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| | location = Dagstuhl, Germany
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| | publisher = [[Dagstuhl|Schloss Dagstuhl – Leibniz-Zentrum für Informatik]]
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| | series = Dagstuhl Seminar Proceedings
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| | title = Parameterized Complexity and Approximation Algorithms
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| | url = http://drops.dagstuhl.de/opus/volltexte/2010/2499
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| | year = 2010}}.
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| *{{citation|first=P. J.|last=Slater|contribution=Leaves of trees|title=Proc. 6th Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975)|series=Congressus Numerantium|volume=14|pages=549–559|year=1975|publisher=Utilitas Math.|location=Winnipeg|mr=0422062}}.
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| *{{citation|first=P. J.|last=Slater|title=Dominating and reference sets in a graph|journal=Journal of Mathematical and Physical Sciences|volume=22|issue=4|pages=445–455|year=1988|mr=0966610}}.
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| [[Category:Graph invariants]]
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