Abstract
For an undirected graph G = (V,E), we say that for ℓ,u,v ∈ V, ℓ separates u from v if the distance between u and ℓ differs from the distance from v to ℓ. A set of vertices L ⊆ V is a feasible solution if for every pair of vertices u,v ∈ V there is ℓ ∈ L that separates u from v. The metric dimension of a graph is the minimum cardinality of such a feasible solution. Here, we extend this well-studied problem to the case where each vertex v has a non-negative cost, and the goal is to find a feasible solution with a minimum total cost. This weighted version is NP-hard since the unweighted variant is known to be NP-hard. We show polynomial time algorithms for the cases where G is a path, a tree, a cycle, a cograph, a k-edge-augmented tree (that is, a tree with additional k edges) for a constant value of k, and a (not necessarily complete) wheel. The results for paths, trees, cycles, and complete wheels extend known polynomial time algorithms for the unweighted version, whereas the other results are the first known polynomial time algorithms for these classes of graphs even for the unweighted version. Next, we extend the set of graph classes for which computing the unweighted metric dimension of a graph is known to be NPC by showing that for split graphs, bipartite graphs, co-bipartite graphs, and line graphs of bipartite graphs, the problem of computing the unweighted metric dimension of the graph is NPC.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Babai, L.: On the order of uniprimitive permutation groups. Annals of Mathematics 113(3), 553–568
Beerliova, Z., Eberhard, F., Erlebach, T., Hall, A., Hoffmann, M., Mihalák, M., Ram, L.S.: Network discovery and verification. IEEE Journal on Selected Areas in Communications 24(12), 2168–2181 (2006)
Cáceres, J., Hernando, M.C., Mora, M., Pelayo, I.M., Puertas, M.L., Seara, C., Wood, D.R.: On the metric dimension of cartesian products of graphs. SIAM Journal on Discrete Mathematics 21(2), 423–441 (2007)
Chartrand, G., Eroh, L., Johnson, M.A., Oellermann, O.R.: Resolvability in graphs and the metric dimension of a graph. Discrete Applied Mathematics 105(1-3), 99–113 (2000)
Chartrand, G., Zhang, P.: The theory and applications of resolvability in graphs: A survey. Congressus Numerantium 160, 47–68 (2003)
Chvátal, V.: Mastermind. Combinatorica 3(3), 325–329 (1983)
Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM Journal on Computing 14(4), 926–934 (1985)
Diaz, J., Pottonen, O., Serna, M., van Leeuwen, E.J.: On the complexity of metric dimension. CoRR, abs/1107.2256. Proc. of ESA 2012 (to appear, 2012)
Harary, F., Melter, R.A.: The metric dimension of a graph. Ars Combinatoria 2, 191–195 (1976)
Hauptmann, M., Schmied, R., Viehmann, C.: Approximation complexity of metric dimension problem. Journal of Discrete Algorithms (2011) (to appear)
Khuller, S., Raghavachari, B., Rosenfeld, A.: Landmarks in graphs. Discrete Applied Mathematics 70(3), 217–229 (1996)
Melter, R.A., Tomescu, I.: Metric bases in digital geometry. Computer Vision, Graphics, and Image Processing 25, 113–121 (1984)
Sebö, A., Tannier, E.: On metric generators of graphs. Mathematics of Operations Research 29(2), 383–393 (2004)
Shanmukha, B., Sooryanarayana, B., Harinath, K.S.: Metric dimension of wheels. Far East Journal of Applied Mathematics 8(3), 217–229 (2002)
Slater, P.J.: Leaves of trees. Congressus Numerantium 14, 549–559 (1975)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Epstein, L., Levin, A., Woeginger, G.J. (2012). The (Weighted) Metric Dimension of Graphs: Hard and Easy Cases. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2012. Lecture Notes in Computer Science, vol 7551. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34611-8_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-34611-8_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34610-1
Online ISBN: 978-3-642-34611-8
eBook Packages: Computer ScienceComputer Science (R0)