ABSTRACT
Let https://www.w3.org/1998/Math/MathML" display="inline"> G = ( V , E ) be a simple graph. Let https://www.w3.org/1998/Math/MathML" display="inline"> u , v be any two vertices of G. The circular distance between https://www.w3.org/1998/Math/MathML" display="inline"> u and https://www.w3.org/1998/Math/MathML" display="inline"> v denoted by https://www.w3.org/1998/Math/MathML" display="inline"> D c ( u , v ) and is defined by https://www.w3.org/1998/Math/MathML" display="block"> D c u , v = { D u , v + d u , v if u ≠ v 0 if u = v where https://www.w3.org/1998/Math/MathML" display="inline"> D u , v and https://www.w3.org/1998/Math/MathML" display="inline"> d u , v are detour distance and distance between https://www.w3.org/1998/Math/MathML" display="inline"> u and https://www.w3.org/1998/Math/MathML" display="inline"> v respectively. Let https://www.w3.org/1998/Math/MathML" display="inline"> W = { w 1 , w 2 , . . . , w k } ⊂ V ( G ) and https://www.w3.org/1998/Math/MathML" display="inline"> v ∈ V ( G ) . The representation https://www.w3.org/1998/Math/MathML" display="inline"> cr ( v / W ) of https://www.w3.org/1998/Math/MathML" display="inline"> v with respect to https://www.w3.org/1998/Math/MathML" display="inline"> W is the https://www.w3.org/1998/Math/MathML" display="inline"> k -tuple https://www.w3.org/1998/Math/MathML" display="inline"> D c v , w 1 , D c v , w 2 , . . . , D c v , w k . If various vertices of https://www.w3.org/1998/Math/MathML" display="inline"> G have distinct representations with regard to https://www.w3.org/1998/Math/MathML" display="inline"> W , then https://www.w3.org/1998/Math/MathML" display="inline"> W is referred to as a circular resolving set. For each given https://www.w3.org/1998/Math/MathML" display="inline"> G , a circular resolving set of minimum cardinality is referred to as a https://www.w3.org/1998/Math/MathML" display="inline"> cdim -set. The circular metric dimension of https://www.w3.org/1998/Math/MathML" display="inline"> G , denoted by https://www.w3.org/1998/Math/MathML" display="inline"> cdim G , is the cardinality of the https://www.w3.org/1998/Math/MathML" display="inline"> 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 https://www.w3.org/1998/Math/MathML" display="inline"> n ≥ 2 with dimension 1 in the circular metric. It is shown that for every pair of integers https://www.w3.org/1998/Math/MathML" display="inline"> a nd https://www.w3.org/1998/Math/MathML" display="inline"> n with https://www.w3.org/1998/Math/MathML" display="inline"> 1 ≤ a ≤ n − 1 , there exists a connected graph of order https://www.w3.org/1998/Math/MathML" display="inline"> n such that https://www.w3.org/1998/Math/MathML" display="inline"> cdim G = 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.
AMS Subject Classification: 05C12
