Metric dimension (graph theory)
In
Detailed definition
For an ordered subset of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple , 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 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.
Trees
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. 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.[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]
Properties
In Chartrand et al. (2000), it is proved that:
- The metric dimension of a graph G is 1 if and only if G is a path.
- The metric dimension of an n-vertex graph is n − 1 if and only if it is a complete graph.
- The metric dimension of an n-vertex graph is n − 2 if and only if the graph is a complete bipartite graph Ks, t, a split graph , or .
Relations between the order, the metric dimension and the diameter
Khuller, Raghavachari & Rosenfeld (1996) prove the inequality for any n-vertex graph with
For specific graph classes, smaller bounds can hold. For example, Beaudou et al. (2018) proved that for trees (the bound being tight for even values of D), and a bound of the form for outerplanar graphs. The same authors proved that for graphs with no complete graph of order t as a minor and also gave bounds for chordal graphs and graphs of bounded treewidth. The authors Foucaud et al. (2017a) proved bounds of the form for interval graphs and permutation graphs, and bounds of the form for
Computational complexity
Decision complexity
Deciding whether the metric dimension of a graph is at most a given integer is NP-complete.
For any fixed constant k, the graphs of metric dimension at most k can be recognized in
Deciding whether the metric dimension of a tree is at most a given integer can be done in linear time
Approximation complexity
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, .[16] This approximation ratio is close to best possible, as under standard complexity-theoretic assumptions a ratio of 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.[17]
References
Notes
- ^ Slater 1975; Harary & Melter 1976; Khuller, Raghavachari & Rosenfeld 1996. Note Slater's nonstandard definition of the leaves of a tree.
- ^ Feng, Xu & Wang 2013.
- ^ Garey & Johnson 1979.
- ^ a b c Epstein, Levin & Woeginger 2012.
- ^ Hoffmann & Wanke 2013.
- ^ a b Foucaud et al. 2017b.
- ^ Li & Pilipczuk 2022.
- ^ a b c Belmonte et al. 2015.
- ^ Slater 1975; Harary & Melter 1976.
- ^ Fernau et al. 2015.
- ^ Hoffmann, Elterman & Wanke 2016.
- ^ a b Hartung & Nichterlein 2013.
- ^ Eppstein 2015.
- ^ Khuller, Raghavachari & Rosenfeld 1996.
- ^ a b Hauptmann, Schmied & Viehmann 2012.
- ^ Hartung 2014.
Bibliography
- Beaudou, Laurent; Dankelmann, Peter; Foucaud, Florent; Henning, Michael A.; Mary, Arnaud; Parreau, Aline (2018), "Bounding the order of a graph using its diameter and metric dimension: a study through tree decompositions and VC dimension", SIAM Journal on Discrete Mathematics, 32 (2): 902–918, S2CID 51882750
- Belmonte, R.; Fomin, F. V.; Golovach, P. A.; Ramanujan, M. S. (2015), "Metric dimension of bounded width graphs", in Italiano, G. F.; Pighizzini, G.; Sannella, D. T. (eds.), Mathematical Foundations of Computer Science 2015 – MFCS 2015: 40th International Symposium, Milan, Italy, August 24-28, 2015, Proceedings, ISBN 978-3-662-48053-3.
- Blumenthal, L.M. (1953), Theory and Applications of Distance Geometry, Clarendon, Oxford.
- Bonnet, E.; Purohit, N. (2019), "Metric Dimension Parameterized by Treewidth", in Jansen, B. M. P.; Telle, J. A. (eds.), Parameterized and Exact Computation 2019 – IPEC 2019: 14th International Symposium, Proceedings, Leibniz International Proceedings in Informatics (LIPIcs), vol. 148, Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, pp. 5:1–5:15, .
- Buczkowski, P.; S2CID 33390310.
- MR 1780464.
- Díaz, J.; Pottonen, O.; ISBN 978-3-642-33089-6.
- S2CID 1318601.
- Epstein, Leah; Levin, Asaf; ISBN 978-3-642-34610-1.
- Feng, Min; Xu, Min; Wang, Kaishun (2013), "On the metric dimension of line graphs", S2CID 36010185.
- Fernau, Henning; .
- Foucaud, Florent; Mertzios, George B.; Naserasr, Reza; Parreau, Aline; Valicov, Petru (2017a), "Identification, location-domination and metric dimension on interval and permutation graphs. I. Bounds", Theoretical Computer Science, 68: 43–58, S2CID 25244200
- Foucaud, Florent; Mertzios, George B.; Naserasr, Reza; Parreau, Aline; Valicov, Petru (2017b), "Identification, location-domination and metric dimension on interval and permutation graphs. II. Algorithms and complexity", S2CID 1520161.
- ISBN 0-7167-1045-5A1.5: GT61, p. 204.
- MR 0457289.
- Hartung, Sepp (2014), Exploring parameter spaces in coping with computational intractability, PhD thesis, Technische Universität Berlin, retrieved 2015-09-15.
- Hartung, Sepp; Nichterlein, André (2013), "On the parameterized and approximation hardness of metric dimension", 2013 IEEE Conference on Computational Complexity (CCC), Stanford, CA, USA, June 5-7, 2013, Proceedings, IEEE, pp. 266–276, S2CID 684505.
- Hauptmann, Mathias; Schmied, Richard; Viehmann, Claus (2012), "Approximation complexity of metric dimension problem", Journal of Discrete Algorithms, 14: 214–222, MR 2922072.
- Hernando, Carmen; Mora, Mercè; Pelayo, Ignacio M.; Seara, Carlos; hdl:2117/8261.
- Hoffmann, Stefan; Elterman, Alina; Wanke, Egon (2016), "A linear time algorithm for metric dimension of cactus block graphs", Theoretical Computer Science, 630: 43–62,
- Hoffmann, Stefan; Wanke, Egon (2013), "Metric Dimension for Gabriel Unit Disk Graphs Is NP-Complete", in Bar-Noy, Amotz; Halldórsson, Magnús M. (eds.), Algorithms for Sensor Systems: 8th International Symposium on Algorithms for Sensor Systems, Wireless Ad Hoc Networks and Autonomous Mobile Entities, ALGOSENSORS 2012, Ljubljana, Slovenia, September 13-14, 2012, Revised Selected Papers, Lecture Notes in Computer Science, vol. 7718, Springer, pp. 90–92, S2CID 9740623.
- .
- Li, Shaohua; Pilipczuk, Marcin (July 2022), "Hardness of metric dimension in graphs of constant treewidth", Algorithmica, 84 (11): 3110–3155, S2CID 231979414
- Lokshtanov, Daniel (2010), "Open problems – Parameterized complexity and approximation algorithms: Metric Dimension", in Demaine, Erik D.; Hajiaghayi, MohammadTaghi; Marx, Dániel (eds.), Parameterized Complexity and Approximation Algorithms, Dagstuhl Seminar Proceedings, Dagstuhl, Germany: Schloss Dagstuhl – Leibniz-Zentrum für Informatik.
- Slater, P. J. (1975), "Leaves of trees", Proc. 6th Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), Congressus Numerantium, vol. 14, Winnipeg: Utilitas Math., pp. 549–559, MR 0422062.
- Slater, P. J. (1988), "Dominating and reference sets in a graph", Journal of Mathematical and Physical Sciences, 22 (4): 445–455, MR 0966610.