Abstract
A vertex k ∈ VG determined two elements (vertices or edges) 1, m ∈ VG U EG, if dG(k, 1) =6 dG(k, m). A setRm of vertices in a graph G is a mixed metric generator for G, if two distinct elements (vertices or edges) are determined by some vertex set of Rm. The least number of elements in the vertex set of Rm is known as mixed metric dimension, and denoted as dimm(G). In this article, the mixed metric dimension of some path related graphs is obtained. Those path related graphs are P2n the square of a path, T(Pn) total graph of a path, the middle graph of a path M(Pn), and splitting graph of a path S(Pn). We proved that these families of graphs have constant and unbounded mixed metric dimension, respectively. We further presented an improved result for the metric dimension of the splitting graph of a path S(Pn).
Highlights
When some vertex set of a graph resolves the graphs’ vertices, the authors called it the metric dimension
AND PRELIMINARY RESULTSLet for a graph G = (VG, EG), where VG, interpret the vertices, and EG the edges of a graph
While discussing the mixed metric dimension structure, it is imperative to note that a graph with a pendant vertex cannot form a mixed metric generator
Summary
When some vertex set of a graph resolves the graphs’ vertices, the authors called it the metric dimension. The invariant of mixed metric dimension is studied for some general families of graphs, as mentioned in the following results. Remark 1: The graphs studied in this article are related to the path graphs, so the mixed metric generator for these families of graphs must contain both end-vertices. Let Rm = {x1, x2, xn−1, xn} be the mixed resolving set for the square of a path graphs P2n.
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