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Mixed metric dimension of some graphs

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  • Nie, Kairui
  • Xu, Kexiang

Abstract

A vertex gv in a connected graph G is said to distinguish two distinct elements p,q∈V(G)⋃E(G) if dG(p,gv)≠dG(q,gv). A subset W⊆V(G) is a mixed metric generator of G if every two distinct elements from V(G)⋃E(G) are distinguished by W. The mixed metric dimension of G, denoted by βm(G), is the minimum cardinality of mixed metric generators in it. In this work, we first answer the problem of characterizing graphs G which achieve βm(G)=n−g(G)+3 in [Appl. Math. Comput. 314 (2017) 429–438] where g(G) is the girth of G, and then determine bounds on the mixed metric dimension of Cartesian product G□H which resolves two open problems in [Discrete Math. 341 (2018) 2083–2088] as a natural corollary. In addition, we provide two closed formulae for βm(T□Pn) in terms of βm(T) which generalize the result on βm(Ps□Pn) in [Appl. Math. Comput. 314 (2017) 429–438].

Suggested Citation

  • Nie, Kairui & Xu, Kexiang, 2023. "Mixed metric dimension of some graphs," Applied Mathematics and Computation, Elsevier, vol. 442(C).
    • Handle: RePEc:eee:apmaco:v:442:y:2023:i:c:s0096300322008050
    DOI: 10.1016/j.amc.2022.127737
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    References listed on IDEAS

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    1. Sedlar, Jelena & Škrekovski, Riste, 2021. "Extremal mixed metric dimension with respect to the cyclomatic number," Applied Mathematics and Computation, Elsevier, vol. 404(C).
    2. Yuezhong Zhang & Suogang Gao, 2020. "On the edge metric dimension of convex polytopes and its related graphs," Journal of Combinatorial Optimization, Springer, vol. 39(2), pages 334-350, February.
    3. Kelenc, Aleksander & Kuziak, Dorota & Taranenko, Andrej & G. Yero, Ismael, 2017. "Mixed metric dimension of graphs," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 429-438.
    4. Meiqin Wei & Jun Yue & Lily Chen, 2022. "The effect of vertex and edge deletion on the edge metric dimension of graphs," Journal of Combinatorial Optimization, Springer, vol. 44(1), pages 331-342, August.
    5. Wu, Jian & Wang, Li & Yang, Weihua, 2022. "Learning to compute the metric dimension of graphs," Applied Mathematics and Computation, Elsevier, vol. 432(C).
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