Abstract
A family $\mathcal {G}$ of connected graphs is a family with unbounded metric dimension if $dim (\text{G})$ is not constant and depends on the order of graph. In this paper, we compute the metric dimension of the splitting graphs $S(P_{n})$ and $S(C_{n})$ of a path and cycle. We prove that the metric dimension of these graphs varies and depends on the number of vertices of the graph.
Highlights
Graph theory has been used to study the various concepts of navigation in an arbitrary space
The minimum number of machines required to locate each and every vertex of the network is termed as metric dimension and the set of all minimum possible number of landmarks constitute metric basis
The graph obtained is called the splitting graph of graph G, denoted by S(G)
Summary
Graph theory has been used to study the various concepts of navigation in an arbitrary space. A work place can be denoted as a vertex in a graph, and edges denote the connections between places. The problem of minimum machine (or Robots) to be placed at certain nodes to trace each and every node exactly once is worth investigating. The problem can be explained using networks where places are interconnected in which, a navigating agent moves from one vertex to another in the network. The places or vertices of a network where we place the machines (robots) are called landmarks. The minimum number of machines required to locate each and every vertex of the network is termed as metric dimension and the set of all minimum possible number of landmarks constitute metric basis
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