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<p>An open-independent dominating set (OIND-set) <i>S</i> for a graph <i>G</i> is a set of vertices where no vertex in <i>S</i> has more than one neighbor within <i>S</i>, and every vertex in <i>G</i> is dominated by <i>S</i>. The minimum cardinality of an OIND-set is denoted by <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44425_2025_34_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">\(\gamma _{oind}(G)\)</EquationSource> </InlineEquation>. This work presents results concerning the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44425_2025_34_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">\(\gamma _{oind}\)</EquationSource> </InlineEquation> of the lexicographic product <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44425_2025_34_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(G \circ H\)</EquationSource> </InlineEquation>. Specifically, we establish exact values under specific constraints on <i>G</i>, and provide bounds relative to another domination variant, as well as <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44425_2025_34_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">\(\gamma _{oind}(G)\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44425_2025_34_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">\(\gamma _{oind}(H)\)</EquationSource> </InlineEquation>.</p>

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Open-Independent Dominating Sets in Lexicographic Product of Graphs

  • 期刊论文

  • 发表日期：

    2025年10月15日

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  • Erika M. M. Coelho,
  • Lauane M. O. Moraes

摘要

An open-independent dominating set (OIND-set) S for a graph G is a set of vertices where no vertex in S has more than one neighbor within S, and every vertex in G is dominated by S. The minimum cardinality of an OIND-set is denoted by \(\gamma _{oind}(G)\) . This work presents results concerning the \(\gamma _{oind}\) of the lexicographic product \(G \circ H\) . Specifically, we establish exact values under specific constraints on G, and provide bounds relative to another domination variant, as well as \(\gamma _{oind}(G)\) and \(\gamma _{oind}(H)\) .