Resolving Set of Grid Graph and its Variants

Abstract

If the distance in equation 𝑑𝐺 (𝑥, 𝑧) ≠ (𝑦, 𝑧) is valid for separate vertices 𝑥, 𝑦, 𝑧 in graph 𝐺 = (𝑉, � �), we say the pair of vertices 𝑥, 𝑦 can be discriminated by vertex 𝑧. If any two distinct vertices in graph G can be identified by at least one vertex in the vertex subset 𝑆 ⊂ 𝑉, then 𝑆 ⊂ 𝑉 is a resolving set of graph G. The one with the smallest number of nodes among all of the resolving sets is termed a metric basis of graph G, and its cardinality is called the metric dimension of graph G [1]. Let (𝑉, 𝐸) be a simple connected graph. We define the distance between an edge 𝑒 = 𝑥𝑦 and vertex 𝑣 as follows: � �(𝑒, 𝑣) = 𝑚𝑖𝑛{𝑑(𝑥, 𝑣), 𝑑(𝑦, 𝑣)}. Two edges 𝑒1 and 𝑒2 are distinguished by a vertex v if (𝑒1, 𝑣) ≠ (𝑒2, 𝑣). If there is a vertex 𝑠 ∈ 𝑆 such that s differentiates 𝑒1 and 𝑒2 for any two unique edges 𝑒1,2∈ 𝐸, then a set 𝑆 ⊆ 𝑉 is an edge metric generator of a graph (𝑉, 𝐸). An edge basis of G is an edge generating set with the fewest members; the edge dimension of G is the number of elements in an edge basis [2]. This study calculates the diagonally folded grid (DFG) graph's metric and edge metric dimensions.