The Metric Dimension of Some Generalized Petersen Graphs

<jats:p>The distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>d</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> between two distinct vertices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:math> in a graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math> is the length of a shortest <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>-path in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>. For an ordered subset <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mi>W</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">}</mml:mo></mml:math> of vertices and a vertex <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>, the code of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:math> with respect to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M11"><mml:mrow><mml:mi>W</mml:mi></mml:mrow></mml:math> is the ordered <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M12"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>-tuple <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M13"><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>W</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math>. The set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M14"><mml:mrow><mml:mi>W</mml:mi></mml:mrow></mml:math> is a resolving set for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M15"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math> if every two vertices of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M16"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math> have distinct codes. The<jats:italic> metric dimension</jats:italic> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M17"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math> is the minimum cardinality of a resolving set of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M18"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>. In this paper, we first extend the results of the metric dimension of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M19"><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mn mathvariant="normal">3</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M20"><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mn mathvariant="normal">4</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> and study bounds on the metric dimension of the families of the generalized Petersen graphs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M21"><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M22"><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">3</mml:mn><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. The obtained results mean that these families of graphs have constant metric dimension.</jats:p>

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