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https://doi.org/10.1063/1.4994...
Article . 2017 . Peer-reviewed
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On the partition dimension of comb product of path and complete graph

Authors: null Darmaji; Ridho Alfarisi;

On the partition dimension of comb product of path and complete graph

Abstract

For a vertex v of a connected graph G(V, E) with vertex set V(G), edge set E(G) and S ⊆ V(G). Given an ordered partition Π = {S1, S2, S3, …, Sk} of the vertex set V of G, the representation of a vertex v ∈ V with respect to Π is the vector r(v|Π) = (d(v, S1), d(v, S2), …, d(v, Sk)), where d(v, Sk) represents the distance between the vertex v and the set Sk and d(v, Sk) = min{d(v, x)|x ∈ Sk}. A partition Π of V(G) is a resolving partition if different vertices of G have distinct representations, i.e., for every pair of vertices u, v ∈ V(G), r(u|Π) ≠ r(v|Π). The minimum k of Π resolving partition is a partition dimension of G, denoted by pd(G). Finding the partition dimension of G is classified to be a NP-Hard problem. In this paper, we will show that the partition dimension of comb product of path and complete graph. The results show that comb product of complete grapph Km and path Pn namely pd(Km⊳Pn)=m where m ≥ 3 and n ≥ 2 and pd(Pn⊳Km)=m where m ≥ 3, n ≥ 2 and m ≥ n.

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
10
Top 10%
Top 10%
Top 10%
bronze