Dimensi Metrik Lokal pada Operasi Korona Graf Ular Segitiga dengan Graf Lintasan Orde Dua

Abstract

Graphs were first introduced by Leonard Euler through the Königsberg Bridge problem in 1736. Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. The concept of distance in graphs leads to the notions of metric dimension and local metric dimension. Let $W \subset V(G)$ with $W = \{w_1, w_2, \dots, w_n\}$. The representation of a vertex $x \in V(G)$ with respect to $W$ is defined by $r(x \mid W) = (d(x, w_1), d(x, w_2), \dots, d(x, w_n)).$ The set $W$ is called a local resolving set of $G$ if for every pair of adjacent vertices $u, v \in V(G)$, $r(u \mid W) \ne r(v \mid W)$. The minimum cardinality of such a set is called the local metric dimension of $G$ and is denoted by $\dim_{\ell}(G)$. This research aims to determine the metric dimension and local metric dimension of the triangular snake graph $T_n$, as well as graphs obtained from the corona operation between $T_n$ and a path graph of order two. The method used is a literature study with an analysis of graph structure and vertex distances. The results show that both the metric dimension and the local metric dimension of the triangular snake graph are equal to $2$. Moreover, the local metric dimension of $T_n \odot P_2$ is $2n+1$, while that of $P_2 \odot T_n$ is $n+3$ for odd $n$ and $n+2$ for even $n$.