On Partition Dimension of Some Cycle-Related Graphs

Abstract

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

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