The metric-locating-dominating bound for doubly resolving sets

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Let GG be a graph. Write dim(G){\rm \dim}(G) for its metric dimension, γ(G)\gamma(G) for its domination number, and ψ(G)\psi(G) for its doubly resolving number. Metric-locating-dominating bound. For every graph GG, it holds that

ψ(G)dim(G)+γ(G).\psi(G)\leq {\rm \dim}(G)+\gamma(G).

This conjecture is supported by the inequality γM(G)dim(G)+γ(G)\gamma_M(G)\leq {\rm \dim}(G)+\gamma(G), where γM(G)\gamma_M(G) denotes the metric-locating-dominating number. The preceding results establish related bounds, including ψ(G)2γM(G)\psi(G)\leq 2\gamma_M(G), but the stated inequality is not resolved here.

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Primary source

Antonio González, Carmen Hernando and Mercè Mora, “New results on metric-locating-dominating sets of graphs”, arXiv:1604.03861 (2016).

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