Constant Time Calculation of the Metric Dimension of the Join of Path Graphs

Abstract

The distance between two vertices of a simple connected graph G, denoted as d(u,v), is the length of the shortest path from u to v and is always symmetrical. An ordered subset W=w1,w2,w3,⋯,wk of V(G) is a resolving set for G, if for ∀u,v∈V(G), there exists wi∈W ∋ d(u,wi)≠d(v,wi). A resolving set with minimal cardinality is called the metric basis. The metric dimension of G is the cardinality of metric basis of G and is denoted as dim(G). For the graph G1=(V1,E1,) and G2=(V2,E2), their join is denoted by G1+G2. The vertex set of G1+G2 is V1∪V2 and the edge set is E=E1∪E2∪uv,u∈V1,v∈V2. In this article, we show that the metric dimension of the join of two path graphs is unbounded because of its dependence on the size of the paths. We also provide a general formula to determine this metric dimension. We also develop algorithms to obtain metric dimensions and a metric basis for the join of path graphs, with respect to its symmetries.

The Secure Metric Dimension of the Globe Graph and the Flag Graph

“…Te metric dimension of a graph is an NP-hard problem, as was mentioned in [32]. Furthermore, business networks, chemical structures, wireless communication networks, and electrical networks [33][34][35][36][37][38][39][40][41][42][43][44] all make use of secure metric dimension theory.…”

Connected metric dimension of the class of ladder graphs

“…According to [2], the connected metric dimension of the wheel graph 𝑊 , 𝑛 ≥ 7, is + 1, while it is for the Petersen graph 𝑃 is 4, and if 𝑣 is an end vertex of the tree 𝑇, then it is at a vertex of 𝑇; otherwise, it is at a vertex of 𝑇 that is 2. Further information might be found in the literature [16][17][18][19][20][21][22], and some future notions may be applied to some applications like [23,24].…”