Dimensi Metrik Monofonik Lokal Graf Degree Splitting

Satriadi, Muhammad Yusti Permana (2024) Dimensi Metrik Monofonik Lokal Graf Degree Splitting. Other thesis, Institut Teknologi Sepuluh Nopember.

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Abstract

Misalkan G = (V, E) adalah graf terhubung dengan himpunan simpul V dan himpunan sisi E, W = {v1, v2, ..., vk} ⊂ V adalah himpunan terurut, dan u, v ∈ V . Representasi monofonik mr(v|W) dari v terhadap W adalah k-tuple
(dm(v, v1), dm(v, v2), ..., dm(v, vk)), dengan dm(v, vi), i ∈ {1, 2, .., k} adalah jarak monofonik dari simpul v ke simpul vi. Himpunan W disebut himpunan pembeda
monofonik dari G jika mr(u|W) ̸= mr(v|W). Himpunan W disebut himpunan pembeda monofonik lokal jika simpul-simpul yang bertetangga dari G memiliki representasi monofonik yang berbeda terhadap W. Himpunan pembeda monofonik lokal dengan
banyaknya elemen minimal disebut himpunan pembeda monofonik lokal minimal untuk G, dan kardinalitasnya disebut dimensi metrik monofonik lokal dari G, mdiml(G). Graf degree splitting adalah graf dengan V = S1∪S2∪S3∪...∪Sk∪T dan T = V − Sj Sj, dengan setiap Sj adalah himpunan simpul berderajat sama dan banyak elemennya minimal dua. Graf degree splitting dari G, DS(G), diperoleh dari G dengan menambahkan simpulsimpul w1, w2, ..., wj dan menghubungkan wj ke setiap simpul di Sj untuk 1 ≤ j ≤ k, dengan k banyaknya derajat simpul-simpul di G. Dalam Tugas Akhir ini ditentukan dan dianalisis dimensi metrik monofonik lokal dari graf-graf khusus dan graf degree splitting. Dari penelitian ini diperoleh: mdiml(Pn) = 1; mdiml(Cn) = 1 jika n genap dan mdiml(Cn) = 2 jika n ganjil; mdiml(Sn) = 1; mdiml(Kn) = n − 1; mdiml(Km,n) = 1; mdiml(DS(Pn)) = 1 untuk n ganjil kecuali 5 dan mdiml(DS(Pn)) = 2 jika n genap
atau n = 5; mdiml(DS(Cn)) = 2; mdiml(DS(Kn)) = n; mdiml(DS(Sn)) = 1; mdiml(DS(Km,n) = 1 jika m ̸= n dan mdiml(DS(Km,n) = 2 jika m =n.
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Let G = (V, E) be a connected graph with vertex set V and edge set E, W = {v1, v2, . . . , vk} ⊂ V is an ordered set, and u, v ∈ V . The monophonic representation mr(v|W) of v with respect to W is the k-tuple (dm(v, v1), dm(v, v2), . . . , dm(v, vk)), where
dm(v, vi), i ∈ {1, 2, . . . , k}, is the monophonic distance from vertex v to vertex vi . The set
W is called a monophonic resolving set of G if mr(u|W) ̸= mr(v|W). The set W is called a local monophonic resolving set if the adjacent vertices of G have different monophonic
representations with respect to W. A local monophonic resolving set with the minimum number of elements is called a minimal local monophonic resolving set for G, and its
cardinality is called the local monophonic metric dimension of G, denoted by mdiml(G). A degree splitting graph is a graph with V = S1∪S2∪S3∪. . .∪Sk ∪T and T = V −
S j Sj, where each Sj is a set of vertices with the same degree and has at least two elements. The degree splitting graph of G, denoted by DS(G), is obtained from G by adding vertices w1, w2, . . . , wj and connecting wj to every vertex in Sj for 1 ≤ j ≤ k, where k is the number of degrees of the vertices in G. In this Thesis, the local monophonic metric dimension of special graphs and degree splitting graphs are determined and analyzed. The results of this research are as follows: mdiml(Pn) = 1; mdiml(Cn) = 1 if n is even and mdiml(Cn) = 2 if n is odd; mdiml(Sn) = 1; mdiml(Kn) = n − 1; mdiml(Km,n) = 1; mdiml(DS(Pn)) = 1 for odd n except 5 and mdiml(DS(Pn)) = 2 if n is even or n = 5;
mdiml(DS(Cn)) = 2; mdiml(DS(Kn)) = n; mdiml(DS(Sn)) = 1; mdiml(DS(Km,n) = 1 if m ̸= n and mdiml(DS(Km,n) = 2 if m = n.

Item Type:	Thesis (Other)
Uncontrolled Keywords:	dimensi metrik, dimensi metrik lokal, dimensi metrik monofonik, dimensi, metrik monofonik lokal, graf degree splitting
Subjects:	Q Science > QA Mathematics > QA159 Algebra Q Science > QA Mathematics > QA166 Graph theory Q Science > QA Mathematics > QA640.7 Discrete geometry Q Science > QA Mathematics > QA76.87 Neural networks (Computer Science)
Divisions:	Faculty of Science and Data Analytics (SCIENTICS) > Mathematics > 44201-(S1) Undergraduate Thesis
Depositing User:	Muhammad Yusti Permana Satriadi
Date Deposited:	07 Aug 2024 03:47
Last Modified:	28 Aug 2024 08:01
URI:	http://repository.its.ac.id/id/eprint/114387

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