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A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs

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Abstract

The metric dimension dim(G) of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z(G) of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V(G)S are colored white) such that V(G) is turned black after finitely many applications of “the color-change rule”: a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim(T) ≤ Z(T) for a tree T, and that dim(G) ≤ Z(G)+1 if G is a unicyclic graph; along the way, we characterize trees T attaining dim(T) = Z(T). For a general graph G, we introduce the “cycle rank conjecture”. We conclude with a proof of dim(T) − 2 ≤ dim(T + e) ≤ dim(T) + 1 for \(e \in E\left( {\bar T} \right)\).

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Acknowledgements

The authors wish to thank the anonymous referees for their comments and suggestions.

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Authors and Affiliations

  1. University of Wisconsin Oshkosh, Oshkosh, WI, 54901, USA

    Linda Eroh

  2. Texas A & M University at Galveston, Galveston, TX, 77553, USA

    Cong X. Kang & Eunjeong Yi

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  1. Linda Eroh
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  2. Cong X. Kang
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  3. Eunjeong Yi
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Correspondence to Eunjeong Yi.

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Eroh, L., Kang, C.X. & Yi, E. A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs. Acta. Math. Sin.-English Ser. 33, 731–747 (2017). https://doi.org/10.1007/s10114-017-4699-4

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