Dimensi Metrik Ketetanggaan Aman Graf

Afara, Najmi Nur (2025) Dimensi Metrik Ketetanggaan Aman Graf. Other thesis, Institut Teknologi Sepuluh Nopember.

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Abstract

Dimensi metrik ketetanggaan aman graf merupakan penggabungan konsep dari dimensi metrik dan dimensi metrik aman graf. Dalam menggabungkan konsep tersebut, perlu memahami dimensi metrik ketetanggaan graf, yang penentuan basisnya memperhatikan penempatan urutan simpul, sedangkan pada dimensi metrik aman, penentuan basisnya memperhatikan semua basis yang berlaku untuk setiap simpul. Diberikan graf terhubung G dan himpunan terurut W yang merupakan himpunan bagian dari V(G) = {v₁, v₂, ..., vₙ}. Untuk setiap simpul v dalam V(G), terdapat representasi ketetanggaan dari v terhadap W, yaitu k-vektor r_A(v|W) = (d_A(v, w₁), d_A(v, w₂), ..., d_A(v, wₖ), dengan jarak ketetanggaan d_A(v, wᵢ) didefinisikan sebagai berikut: d_A(v, wᵢ) = 0 jika v = wᵢ, d_A(v, wᵢ) = 1 jika v terhubung langsung dengan wᵢ, dan d_A(v, wᵢ) = 2 jika v tidak terhubung dengan wᵢ. Himpunan W disebut himpunan pembeda ketetanggaan aman dari G jika vektor r_A(v|W) tidak sama dengan r_A(w|W) untuk setiap v dan w dalam V(G), sehingga himpunan tersebut merupakan pembeda ketetanggaan dari G. Selain itu, untuk semua x dalam V(G) - W, terdapat y dalam W sedemikian hingga (W - {y}) ∪ {x} merupakan himpunan pembeda ketetanggaan dari G. Kardinalitas minimal dari himpunan pembeda ketetanggaan aman W disebut dimensi metrik ketetanggaan aman dari G dan dinotasikan dengan sdim_A(G). Pada penelitian ini, dihasilkan dimensi metrik ketetanggaan graf diamonwork {Cₙ}^m dan dimensi metrik ketetanggaan aman graf G dalam {Pₙ, Cₙ, Sₙ, Kₙ, K_{r,s}, {Cₙ}^m}.
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The metric dimension of a secure adjacency metric dimension of a graph is a combination of the concepts of adjacency metric dimension and secure metric dimension of graphs. In merging these concepts, it is essential to understand the adjacency metric dimension of graphs, which involves determining a basis for the graph while considering the placement of the vertex order. In contrast, the secure metric dimension focuses on determining a basis for the graph while accounting for all bases applicable to each vertex. Given a connected graph G and an ordered set W which is a subset of V(G) = {v₁, v₂, ..., vₙ}, for each vertex v in V(G), the adjacency representation of v with respect to W is defined as the k-vector r_A(v|W) = (d_A(v, w₁), d_A(v, w₂), ..., d_A(v, wₖ), where the distance of adjacency d_A(v, wᵢ) is defined as follows: d_A(v, wᵢ) = 0 if v = wᵢ, d_A(v, wᵢ) = 1 if v is adjacent to wᵢ, and d_A(v, wᵢ) = 2 if v is not adjacent to wᵢ. The set W is called a secure adjacency resolving set for G if the vector r_A(v|W) is not equal to r_A(w|W) for every v and w in V(G). Moreover, for every x in V(G) - W, there exists a y in W such that the set (W - {y}) ∪ {x} is also an adjacency resolving set for G. The minimal cardinality of the secure adjacency resolving set W is referred to as the secure adjacency metric dimension of G, denoted as sdim_A(G). This research results in the determination of the adjacency metric dimension of the graph diamondwork {Cₙ}^m and the secure adjacency metric dimension for graphs G in {Pₙ, Cₙ, Sₙ, Kₙ, K_{r,s}, {Cₙ}^m}.

Item Type:	Thesis (Other)
Uncontrolled Keywords:	Dimensi Metrik, Dimensi Metrik Ketetanggaan, Dimensi Metrik Aman, Dimensi Metrik Ketetanggaan Aman, Graf Diamondwork. Metric Dimension, Adjacency Metric Dimension, Secure Metric Dimension, Secure Adjacency Metric Dimension, Diamondwork Graph
Subjects:	Q Science > QA Mathematics > QA166 Graph theory
Divisions:	Faculty of Mathematics and Science > Mathematics > 44201-(S1) Undergraduate Thesis
Depositing User:	Najmi Nur Afara
Date Deposited:	01 Aug 2025 04:08
Last Modified:	01 Aug 2025 04:08
URI:	http://repository.its.ac.id/id/eprint/125738

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