Dimensi Metrik Ketetanggaan Lokal Aman Graf

Qurratu`ain, Nisrina (2025) Dimensi Metrik Ketetanggaan Lokal Aman Graf. Other thesis, Institut Teknologi Sepuluh Nopember.

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Abstract

Diberikan graf terhubung G dengan himpunan simpul V (G), himpunan sisi E(G) dan u,v∈V(G). Jarak ketetanggaan antara simpul u dan v, dA(u, v). Representasi ketetanggaan simpul v terhadap himpunan simpul terurut W = {w1,w2, ...,wk} ⊆ V (G), rA(v|W) adalah k-tuple (dA(v,w1), dA(v,w2), . . . , dA(v,wk)). Himpunan W disebut himpunan pembeda ketetanggaan dari G jika rA(u|W) ≠ rA(v|W). Himpunan W disebut himpunan pembeda ketetanggaan lokal dari G jika W adalah himpunan pembeda ketetanggaan dan u bertetangga dengan v. Himpunan pembeda ketetanggaan lokal dengan banyak elemen minimum disebut basis ketetanggaan lokal dari G, dan kardinalitas dari basis ketetanggaan lokal dari G disebut dimensi metrik ketetanggaan lokal dari G, dinotasikan dengan dimAl(G). Himpunan pembeda ketetanggaan lokal dari G dikatakan aman jika untuk setiap v∈(V(G)-W), terdapat w∈W sehingga (W-{w})υ{v} merupakan himpunan pembeda ketetanggaan lokal dari G. Himpunan pembeda ketetanggaan lokal aman dari G dengan banyak elemen minimum disebut basis ketetanggaan lokal aman dari G, dan kardinalitas dari basis ketetanggaan lokal aman dari G disebut dimensi metrik ketetanggaan lokal aman dari G, dinotasikan dengan sdimAl(G). Dalam Tugas Akhir ini ditentukan dan dianalisis dimensi metrik ketetanggaan lokal dan dimensi metrik ketetanggaan lokal aman dari graf G∈{Pn,Cn,Kn,Sn,Kmn,Wn,Fn,Cn⊙ K1}. Selain itu, diperoleh hubungan bahwa dimAl(G) ≤ sdimAl(G).
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Given a connected graph G with vertex set V (G), edge set E(G) and u,v∈V(G). The adjacency distance between vertices u and v, dA(u, v). The adjacency representation of vertex v with respect to the ordered vertex set W = {w1,w2, ...,wk} ⊆ V (G), rA(v|W) is a k-tuple (dA(v,w1), dA(v,w2), . . . , dA(v,wk)). A set W is called an adjacency resolving set of G if rA(u|W) ≠ rA(v|W). A set W is called a local adjacency resolving set of G if W is an adjacency resolving set and u is adjacent to v. The set of local adjacency resolving with minimum number of elements is called the local adjacency basis of G, and the cardinality of the local adjacency basis of G is called the local adjacency metric dimension of G, denoted by dimAl(G). The set of local adjacency resolving of G is said to be secure if for every v∈(V(G)-W), there exists a w∈W such that (W-{w})υ{v} is the set of local adjacency resolving of G. The set of secure local adjacency resolving of G with minimum number of elements is called the secure local adjacency basis of G, and the cardinality of the secure local adjacency basis of G is called the secure local adjacency metric dimension of G, denoted by sdimAl(G). In this Final Project, the local adjacency metric dimension and the secure local adjacency metric dimension of the graph G∈{Pn,Cn,Kn,Sn,Kmn,Wn,Fn,Cn⊙ K1} are determined and analyzed. In addition, the relationship is obtained that dimAl(G) ≤ sdimAl(G).

Item Type:	Thesis (Other)
Uncontrolled Keywords:	Dimensi metrik aman, Dimensi metrik ketetanggaan, Dimensi metrik ketetanggaan lokal, Dimensi metrik lokal, Jarak ketetanggaan. Adjacency distance, Adjacency metric dimension, Local adjacency metric dimension, Local metric dimension, Secure metric dimension.
Subjects:	Q Science Q Science > QA Mathematics Q Science > QA Mathematics > QA166 Graph theory
Divisions:	Faculty of Science and Data Analytics (SCIENTICS) > Mathematics > 44201-(S1) Undergraduate Thesis
Depositing User:	Nisrina Qurratu'ain
Date Deposited:	01 Feb 2025 15:31
Last Modified:	01 Feb 2025 15:31
URI:	http://repository.its.ac.id/id/eprint/117450

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