On the local metric dimension of t-fold wheel, Pn o Km, and generalized fan

Let G be a connected graph and let u, v ∈ <e...

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Main Authors:	Rokhana Ayu Solekhah, Tri Atmojo Kusmayadi
Format:	Article
Language:	English
Published:	InaCombS; Universitas Jember; dan Universitas Indonesia 2018-12-01
Series:	Indonesian Journal of Combinatorics
Online Access:	http://www.ijc.or.id/index.php/ijc/article/view/56

id	doaj-30806f65ed594e91829233cbbf6f958b
record_format	Article
spelling	doaj-30806f65ed594e91829233cbbf6f958b2020-11-24T23:44:28ZengInaCombS; Universitas Jember; dan Universitas IndonesiaIndonesian Journal of Combinatorics2541-22052018-12-0122889610.19184/ijc.2018.2.2.420On the local metric dimension of t-fold wheel, Pn o Km, and generalized fanRokhana Ayu Solekhah0Tri Atmojo Kusmayadi1Universitas Sebelas Maret, SurakartaUniversitas Sebelas Maret, Surakarta<p>Let <span class="math"><em>G</em></span> be a connected graph and let <span class="math"><em>u</em>, <em>v</em></span> <span class="math"> ∈ </span> <span class="math"><em>V</em>(<em>G</em>)</span>. For an ordered set <span class="math"><em>W</em> = {<em>w</em><sub>1</sub>, <em>w</em><sub>2</sub>, ..., <em>w</em><sub><em>n</em></sub>}</span> of <span class="math"><em>n</em></span> distinct vertices in <span class="math"><em>G</em></span>, the representation of a vertex <span class="math"><em>v</em></span> of <span class="math"><em>G</em></span> with respect to <span class="math"><em>W</em></span> is the <span class="math"><em>n</em></span>-vector <span class="math"><em>r</em>(<em>v</em>∣<em>W</em>) = (<em>d</em>(<em>v</em>, <em>w</em><sub>1</sub>), <em>d</em>(<em>v</em>, <em>w</em><sub>2</sub>), ..., </span> <span class="math"><em>d</em>(<em>v</em>, <em>w</em><sub><em>n</em></sub>))</span>, where <span class="math"><em>d</em>(<em>v</em>, <em>w</em><sub><em>i</em></sub>)</span> is the distance between <span class="math"><em>v</em></span> and <span class="math"><em>w</em><sub><em>i</em></sub></span> for <span class="math">1 ≤ <em>i</em> ≤ <em>n</em></span>. The set <span class="math"><em>W</em></span> is a local metric set of <span class="math"><em>G</em></span> if <span class="math"><em>r</em>(<em>u</em> ∣ <em>W</em>) ≠ <em>r</em>(<em>v</em> ∣ <em>W</em>)</span> for every pair <span class="math"><em>u</em>, <em>v</em></span> of adjacent vertices of <span class="math"><em>G</em></span>. The local metric set of <span class="math"><em>G</em></span> with minimum cardinality is called a local metric basis for <span class="math"><em>G</em></span> and its cardinality is called a local metric dimension, denoted by <span class="math"><em>l</em><em>m</em><em>d</em>(<em>G</em>)</span>. In this paper we determine the local metric dimension of a <span class="math"><em>t</em></span>-fold wheel graph, <span class="math"><em>P</em><sub><em>n</em></sub></span> <span class="math"> ⊙ </span> <span class="math"><em>K</em><sub><em>m</em></sub></span> graph, and generalized fan graph.</p>http://www.ijc.or.id/index.php/ijc/article/view/56
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Rokhana Ayu Solekhah Tri Atmojo Kusmayadi
spellingShingle	Rokhana Ayu Solekhah Tri Atmojo Kusmayadi On the local metric dimension of t-fold wheel, Pn o Km, and generalized fan Indonesian Journal of Combinatorics
author_facet	Rokhana Ayu Solekhah Tri Atmojo Kusmayadi
author_sort	Rokhana Ayu Solekhah
title	On the local metric dimension of t-fold wheel, Pn o Km, and generalized fan
title_short	On the local metric dimension of t-fold wheel, Pn o Km, and generalized fan
title_full	On the local metric dimension of t-fold wheel, Pn o Km, and generalized fan
title_fullStr	On the local metric dimension of t-fold wheel, Pn o Km, and generalized fan
title_full_unstemmed	On the local metric dimension of t-fold wheel, Pn o Km, and generalized fan
title_sort	on the local metric dimension of t-fold wheel, pn o km, and generalized fan
publisher	InaCombS; Universitas Jember; dan Universitas Indonesia
series	Indonesian Journal of Combinatorics
issn	2541-2205
publishDate	2018-12-01
description	<p>Let <span class="math"><em>G</em></span> be a connected graph and let <span class="math"><em>u</em>, <em>v</em></span> <span class="math"> ∈ </span> <span class="math"><em>V</em>(<em>G</em>)</span>. For an ordered set <span class="math"><em>W</em> = {<em>w</em><sub>1</sub>, <em>w</em><sub>2</sub>, ..., <em>w</em><sub><em>n</em></sub>}</span> of <span class="math"><em>n</em></span> distinct vertices in <span class="math"><em>G</em></span>, the representation of a vertex <span class="math"><em>v</em></span> of <span class="math"><em>G</em></span> with respect to <span class="math"><em>W</em></span> is the <span class="math"><em>n</em></span>-vector <span class="math"><em>r</em>(<em>v</em>∣<em>W</em>) = (<em>d</em>(<em>v</em>, <em>w</em><sub>1</sub>), <em>d</em>(<em>v</em>, <em>w</em><sub>2</sub>), ..., </span> <span class="math"><em>d</em>(<em>v</em>, <em>w</em><sub><em>n</em></sub>))</span>, where <span class="math"><em>d</em>(<em>v</em>, <em>w</em><sub><em>i</em></sub>)</span> is the distance between <span class="math"><em>v</em></span> and <span class="math"><em>w</em><sub><em>i</em></sub></span> for <span class="math">1 ≤ <em>i</em> ≤ <em>n</em></span>. The set <span class="math"><em>W</em></span> is a local metric set of <span class="math"><em>G</em></span> if <span class="math"><em>r</em>(<em>u</em> ∣ <em>W</em>) ≠ <em>r</em>(<em>v</em> ∣ <em>W</em>)</span> for every pair <span class="math"><em>u</em>, <em>v</em></span> of adjacent vertices of <span class="math"><em>G</em></span>. The local metric set of <span class="math"><em>G</em></span> with minimum cardinality is called a local metric basis for <span class="math"><em>G</em></span> and its cardinality is called a local metric dimension, denoted by <span class="math"><em>l</em><em>m</em><em>d</em>(<em>G</em>)</span>. In this paper we determine the local metric dimension of a <span class="math"><em>t</em></span>-fold wheel graph, <span class="math"><em>P</em><sub><em>n</em></sub></span> <span class="math"> ⊙ </span> <span class="math"><em>K</em><sub><em>m</em></sub></span> graph, and generalized fan graph.</p>
url	http://www.ijc.or.id/index.php/ijc/article/view/56
work_keys_str_mv	AT rokhanaayusolekhah onthelocalmetricdimensionoftfoldwheelpnokmandgeneralizedfan AT triatmojokusmayadi onthelocalmetricdimensionoftfoldwheelpnokmandgeneralizedfan
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On the local metric dimension of t-fold wheel, Pn o Km, and generalized fan

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