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

Darmaji*, Ridho Alfarisi

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

13 Citations (Scopus)

## 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.

Original language English International Conference on Mathematics - Pure, Applied and Computation Empowering Engineering using Mathematics Dieky Adzkiya American Institute of Physics Inc. 9780735415478 https://doi.org/10.1063/1.4994441 Published - 1 Aug 2017 2nd International Conference on Mathematics - Pure, Applied and Computation: Empowering Engineering using Mathematics, ICoMPAC 2016 - Surabaya, IndonesiaDuration: 23 Nov 2016 → …

### Publication series

Name AIP Conference Proceedings 1867 0094-243X 1551-7616

### Conference

Conference 2nd International Conference on Mathematics - Pure, Applied and Computation: Empowering Engineering using Mathematics, ICoMPAC 2016 Indonesia Surabaya 23/11/16 → …

## Keywords

• Resolving partition
• comb product
• complete graph
• partition dimension
• path

## Fingerprint

Dive into the research topics of 'On the partition dimension of comb product of path and complete graph'. Together they form a unique fingerprint.