The Vertex-Edge Resolvability of Some Wheel-Related Graphs

<jats:p>A vertex <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>w</mi> <mo>∈</mo> <mi>V</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>H</mi> </mrow> </mfenced> </math> </jats:inline-formula> distinguishes (or resolves) two elements (edges or vertices) <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>a</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>V</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>H</mi> </mrow> </mfenced> <mo>∪</mo> <mi>E</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>H</mi> </mrow> </mfenced> </math> </jats:inline-formula> if <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi>d</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>w</mi> <mo>,</mo> <mi>a</mi> </mrow> </mfenced> <mo>≠</mo> <mi>d</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>w</mi> <mo>,</mo> <mi>z</mi> </mrow> </mfenced> </math> </jats:inline-formula>. A set <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <msub> <mrow> <mi>W</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </math> </jats:inline-formula> of vertices in a nontrivial connected graph <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <mi>H</mi> </math> </jats:inline-formula> is said to be a mixed resolving set for <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <mi>H</mi> </math> </jats:inline-formula> if every two different elements (edges and vertices) of <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7"> <mi>H</mi> </math> </jats:inline-formula> are distinguished by at least one vertex of <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8"> <msub> <mrow> <mi>W</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </math> </jats:inline-formula>. The mixed resolving set with minimum cardinality in <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9"> <mi>H</mi> </math> </jats:inline-formula> is called the mixed metric dimension (vertex-edge resolvability) of <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M10"> <mi>H</mi> </math> </jats:inline-formula> and denoted by <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M11"> <mi>m</mi> <mtext> dim</mtext> <mfenced open="(" close=")" separators="|"> <mrow> <mi>H</mi> </mrow> </mfenced> </math> </jats:inline-formula>. The aim of this research is to determine the mixed metric dimension of some wheel graph subdivisions. We specifically analyze and compare the mixed metric, edge metric, and metric dimensions of the graphs obtained after the wheel graphs’ spoke, cycle, and barycentric subdivisions. We also prove that the mixed resolving sets for some of these graphs are independent.</jats:p>

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