<jats:p>Let <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
<mi>G</mi>
</math>
</jats:inline-formula> be a simple connected graph. Suppose <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
<mi mathvariant="normal">Δ</mi>
<mo>=</mo>
<mfenced open="{" close="}" separators="|">
<mrow>
<msub>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…</mo>
<mo>,</mo>
<msub>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mi>l</mi>
</mrow>
</msub>
</mrow>
</mfenced>
</math>
</jats:inline-formula> an <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
<mi>l</mi>
</math>
</jats:inline-formula>-partition of <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
<mi>V</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<mi>G</mi>
</mrow>
</mfenced>
</math>
</jats:inline-formula>. A partition representation of a vertex <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
<mi>α</mi>
<mtext> w</mtext>
<mo>.</mo>
<mtext>r</mtext>
<mo>.</mo>
<mtext>t </mtext>
<mi mathvariant="normal">Δ</mi>
</math>
</jats:inline-formula> is the <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
<mi>l</mi>
<mo>−</mo>
</math>
</jats:inline-formula>vector <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M7">
<mfenced open="(" close=")" separators="|">
<mrow>
<mi>d</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<mi>α</mi>
<mo>,</mo>
<msub>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mn>1</mn>
</mrow>
</msub>
</mrow>
</mfenced>
<mo>,</mo>
<mi>d</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<mi>α</mi>
<mo>,</mo>
<msub>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mn>2</mn>
</mrow>
</msub>
</mrow>
</mfenced>
<mo>,</mo>
<mo>…</mo>
<mo>,</mo>
<mi>d</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<mi>α</mi>
<mo>,</mo>
<msub>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mi>l</mi>
</mrow>
</msub>
</mrow>
</mfenced>
</mrow>
</mfenced>
</math>
</jats:inline-formula>, denoted by <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M8">
<mi>r</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<mi>α</mi>
<mrow>
<mi>|</mi>
<mi mathvariant="normal">Δ</mi>
</mrow>
</mrow>
</mfenced>
</math>
</jats:inline-formula>. Any partition <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M9">
<mi mathvariant="normal">Δ</mi>
</math>
</jats:inline-formula> is referred as resolving partition if <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M10">
<mo>∀</mo>
<msub>
<mrow>
<mi>α</mi>
</mrow>
<mrow>
<mi>i</mi>
</mrow>
</msub>
<mo>≠</mo>
<msub>
<mrow>
<mi>α</mi>
</mrow>
<mrow>
<mi>j</mi>
</mrow>
</msub>
<mo>∈</mo>
<mi>V</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<mi>G</mi>
</mrow>
</mfenced>
</math>
</jats:inline-formula> such that <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M11">
<mi>r</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<msub>
<mrow>
<mi>α</mi>
</mrow>
<mrow>
<mi>i</mi>
</mrow>
</msub>
<mrow>
<mi>|</mi>
<mi mathvariant="normal">Δ</mi>
</mrow>
</mrow>
</mfenced>
<mo>≠</mo>
<mi>r</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<msub>
<mrow>
<mi>α</mi>
</mrow>
<mrow>
<mi>j</mi>
</mrow>
</msub>
<mrow>
<mi>|</mi>
<mi mathvariant="normal">Δ</mi>
</mrow>
</mrow>
</mfenced>
</math>
</jats:inline-formula>. The smallest integer <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M12">
<mi>l</mi>
</math>
</jats:inline-formula> is referred as the partition dimension <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M13">
<mtext>pd</mtext>
<mfenced open="(" close=")" separators="|">
<mrow>
<mi>G</mi>
</mrow>
</mfenced>
</math>
</jats:inline-formula> of <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M14">
<mi>G</mi>
</math>
</jats:inl
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