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The k-metric dimension

Author

Listed:
  • Ron Adar

    (University of Haifa)

  • Leah Epstein

    (University of Haifa)

Abstract

For an undirected graph $$G=(V,E)$$ G = ( V , E ) , a vertex $$\tau \in V$$ τ ∈ V separates vertices u and v (where $$u,v\in V$$ u , v ∈ V , $$u\ne v$$ u ≠ v ) if their distances to $$\tau $$ τ are not equal. Given an integer parameter $$k \ge 1$$ k ≥ 1 , a set of vertices $$L\subseteq V$$ L ⊆ V is a feasible solution, if for every pair of distinct vertices, u, v, there are at least k distinct vertices $$\tau _{1},\tau _{2},\ldots ,\tau _{k}\in L$$ τ 1 , τ 2 , … , τ k ∈ L , each separating u and v. Such a feasible solution is called a landmark set, and the k-metric dimension of a graph is the minimal cardinality of a landmark set for the parameter k. The case $$k=1$$ k = 1 is a classic problem, where in its weighted version, each vertex v has a non-negative cost, and the goal is to find a landmark set with minimal total cost. We generalize the problem for $$k \ge 2$$ k ≥ 2 , introducing two models, and we seek for solutions to both the weighted version and the unweighted version of this more general problem. In the model of all-pairs (AP), k separations are needed for every pair of distinct vertices of V, while in the non-landmarks model (NL), such separations are required only for pairs of distinct vertices in $$V \setminus L$$ V \ L . We study the weighted and unweighted versions for both models (AP and NL), for path graphs, complete graphs, complete bipartite graphs, and complete wheel graphs, for all values of $$k \ge 2$$ k ≥ 2 . We present algorithms for these cases, thus demonstrating the difference between the two new models, and the differences between the cases $$k=1$$ k = 1 and $$k \ge 2$$ k ≥ 2 .

Suggested Citation

  • Ron Adar & Leah Epstein, 2017. "The k-metric dimension," Journal of Combinatorial Optimization, Springer, vol. 34(1), pages 1-30, July.
  • Handle: RePEc:spr:jcomop:v:34:y:2017:i:1:d:10.1007_s10878-016-0073-1
    DOI: 10.1007/s10878-016-0073-1
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    References listed on IDEAS

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    1. András Sebő & Eric Tannier, 2004. "On Metric Generators of Graphs," Mathematics of Operations Research, INFORMS, vol. 29(2), pages 383-393, May.
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    Cited by:

    1. Klavžar, Sandi & Rahbarnia, Freydoon & Tavakoli, Mostafa, 2021. "Some binary products and integer linear programming for k-metric dimension of graphs," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    2. Alejandro Estrada-Moreno, 2021. "The k -Metric Dimension of a Unicyclic Graph," Mathematics, MDPI, vol. 9(21), pages 1-14, November.

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