On The Local Metric Dimension of Line Graph of Special Graph

Abstract

Let G be a simple, nontrivial, and connected graph. 𝑊 = {𝑤 } is a representation of an ordered set of k distinct vertices in a nontrivial connected graph G. The metric code of a vertex v, where 𝑣 ∈ G, the ordered 𝑟(𝑣|𝑊) = (𝑑 ( 𝑣, 𝑤 1 ) , 𝑑 ( 𝑣, 𝑤 2 ) , . . . , 𝑑 ( 𝑣, 𝑤 𝑘 1 , 𝑤 2 , 𝑤 3 , … , 𝑤 𝑘 ) ) of k-vector is representations of v with respect to W, where 𝑑(𝑣, 𝑤 ) is the distance between the vertices v and w i for 1≤ i ≤k. Furthermore, the set W is called a local resolving set of G if 𝑟 ( 𝑢 | 𝑊 ) ≠ 𝑟(𝑣|𝑊) for every pair u,v of adjacent vertices of G. The local metric dimension ldim(G) is minimum cardinality of W. The local metric dimension exists for every nontrivial connected graph G. In this paper, we study the local metric dimension of line graph of special graphs , namely 𝑖 path, cycle, generalized star, and wheel. The line graph L(G) of a graph G has a vertex for each edge of G, and two vertices in L(G) are adjacent if and only if the corresponding edges in G have a vertex in common.