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Metric-locating-dominating sets of graphs for constructing related subsets of vertices

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  • González, Antonio
  • Hernando, Carmen
  • Mora, Mercè

Abstract

A dominating set S of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distances from the elements of S, and the minimum cardinality of such a set is called the metric-location-domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominating sets to other special sets: resolving sets, dominating sets, locating-dominating sets and doubly resolving sets. We first characterize the extremal trees of the bounds that naturally involve metric-location-domination number, metric dimension and domination number. Then, we prove that there is no polynomial upper bound on the location-domination number in terms of the metric-location-domination number, thus extending a result of Henning and Oellermann. Finally, we show different methods to transform metric-locating-dominating sets into locating-dominating sets and doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them concerning parameters that have not been related so far.

Suggested Citation

  • González, Antonio & Hernando, Carmen & Mora, Mercè, 2018. "Metric-locating-dominating sets of graphs for constructing related subsets of vertices," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 449-456.
    • Handle: RePEc:eee:apmaco:v:332:y:2018:i:c:p:449-456
    DOI: 10.1016/j.amc.2018.03.053
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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.
    2. Mladenović, Nenad & Kratica, Jozef & Kovačević-Vujčić, Vera & Čangalović, Mirjana, 2012. "Variable neighborhood search for metric dimension and minimal doubly resolving set problems," European Journal of Operational Research, Elsevier, vol. 220(2), pages 328-337.
    3. Stephen, Sudeep & Rajan, Bharati & Grigorious, Cyriac & William, Albert, 2015. "Resolving-power dominating sets," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 778-785.
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