Dimensi Metrik Graf Kincir K1+nK3

Abstract

The metric dimension is a fundamental concept in graph theory that utilizes the vector representation of distances between vertices and a subset of vertices in a graph. This concept has broad applications in various fields, such as navigation, network localization, and network design. Let  G(V,E) be a connected graph with order n . A subset L={v₁,v₂,v₃} subset of V(G) is called a resolving set, and the representation of a vertex  v with respect to L is a vector (d(v,v₁), d(v,v₂)..d(v,v_k)) , where d(v,v_i) is the distance between  v and v_i . The metric dimension of G, denoted as dim(G), is the smallest cardinality of  L such that every vertex in G has a unique representation. The windmill graph  K₁+nK₃ is a graph obtained by connecting a vertex  x in the complete graph K₁  to every vertex in n copies of the complete graph K₃. This paper employs a structural analysis method focused on the single vertex in  K₁ and determines the vector representations of all vertices in the windmill graph by analyzing inter-vertex distances. The final result shows that the metric dimension,  dim (K₁+nK₃)= 2n, where n  is an integer grether than 2 .