On bounded partition dimension of different families of convex polytopes with pendant edges
Let ψ=(V,E) be a simple connected graph. The distance between ρ1,ρ2∈V(ψ) is the length of a shortest path between ρ1 and ρ2. Let Γ={Γ1,Γ2,…,Γj} be an ordered partition of the vertices of ψ . Let ρ1∈V(ψ) , and r(ρ1|Γ)={d(ρ1,Γ1),d(ρ1,Γ2),…,d(ρ1,Γj)} be a j -tuple. If the representation...
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| Main Authors: | , , |
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| Format: | Article |
| Published: |
AIMS Press
2021
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| Online Access: | http://psasir.upm.edu.my/id/eprint/94422/ https://www.aimspress.com/article/doi/10.3934/math.2022245 |
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| Institution: | Universiti Putra Malaysia |
| Summary: | Let ψ=(V,E)
be a simple connected graph. The distance between ρ1,ρ2∈V(ψ)
is the length of a shortest path between ρ1
and ρ2.
Let Γ={Γ1,Γ2,…,Γj}
be an ordered partition of the vertices of ψ
. Let ρ1∈V(ψ)
, and r(ρ1|Γ)={d(ρ1,Γ1),d(ρ1,Γ2),…,d(ρ1,Γj)}
be a j
-tuple. If the representation r(ρ1|Γ)
of every ρ1∈V(ψ)
w.r.t. Γ
is unique then Γ
is the resolving partition set of vertices of ψ
. The minimum value of j
in the resolving partition set is known as partition dimension and written as pd(ψ).
The problem of computing exact and constant values of partition dimension is hard so one can compute bound for the partition dimension of a general family of graph. In this paper, we studied partition dimension of the some families of convex polytopes with pendant edge such as RPn
, Dpn
and Qpn
and proved that these graphs have bounded partition dimension. |
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