On the Complexity of Metric Dimension

  • Josep Díaz
  • Olli Pottonen
  • Maria Serna
  • Erik Jan van Leeuwen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7501)


The metric dimension of a graph G is the size of a smallest subset L ⊆ V(G) such that for any x,y ∈ V(G) there is a z ∈ L such that the graph distance between x and z differs from the graph distance between y and z. Even though this notion has been part of the literature for almost 40 years, the computational complexity of determining the metric dimension of a graph is still very unclear. Essentially, we only know the problem to be NP-hard for general graphs, to be polynomial-time solvable on trees, and to have a logn-approximation algorithm for general graphs. In this paper, we show tight complexity boundaries for the Metric Dimension problem. We achieve this by giving two complementary results. First, we show that the Metric Dimension problem on bounded-degree planar graphs is NP-complete. Then, we give a polynomial-time algorithm for determining the metric dimension of outerplanar graphs.


Short Path Planar Graph Bifurcation Point Tree Decomposition Dimension Problem 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Josep Díaz
    • 1
  • Olli Pottonen
    • 1
  • Maria Serna
    • 1
  • Erik Jan van Leeuwen
    • 2
  1. 1.Departament de Llenguatges i Sistemes InformaticsUPCBarcelonaSpain
  2. 2.Dept. Computer, Control, Managm. Eng.Sapienza University of RomeItaly

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