On the circular metric dimension of a graph

Let G ¼ ðV; EÞ be a simple graph. Let u; v be any two vertices of G.
The circular distance between u and v denoted by Dcðu; vÞ and is defined by
Dcðu; vÞ ¼ Dðu; vÞ þ dðu; vÞ if u 6¼ v
0 if u ¼ v
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where Dðu; vÞ and dðu; vÞ are detour distance and distance between u and v respectively. Let
W ¼ w1; w2; ::: ;wk f g � VðGÞ and v 2 VðGÞ. The representation crðv=WÞ of v with respect
to W is the k-tuple Dc v; w1 ð Þ; Dc v; w2 ð Þ; ::: ; Dc v; wk ð ð ÞÞ: If various vertices of G have distinct
representations with regard toW, thenW is referred to as a circular resolving set. For each
given G, a circular resolving set of minimum cardinality is referred to as a cdim -set. The circular
metric dimension of G, denoted by cdimðGÞ; is the cardinality of the cdim -set. A few general
qualities that this idea satisfies are examined. A few common graphs’ circular metric dimensions
are found. We characterise connected graphs of order n � 2 with dimension 1 in the circular
metric. It is shown that for every pair of integers a nd n with 1 � a � n � 1; there exists a
connected graph of order n such that cdimðGÞ ¼ a: The circular metric dimension for the total
graph of paths, the middle graph of paths are determined. Additionally, the circular metric
dimension for the corona products of some graphs are determined.