Let G be a nontrivial connected graph and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$d_G(u,v)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the distance between the vertices u and v in G. Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal{I}\mathcal{D}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">I</mi> <mi mathvariant="script">D</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the set of independent dominating sets of G. A set <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$I\in \mathcal{I}\mathcal{D}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mo>∈</mo> <mi mathvariant="script">I</mi> <mi mathvariant="script">D</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is an independent semitotal dominating set of G if for every vertex <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$v\in I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>v</mi> <mo>∈</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation> there exists a vertex <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$u\in I\setminus \{v\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>∈</mo> <mi>I</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mo stretchy="false">{</mo> <mi>v</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$d_G(u,v)=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. The independent semitotal domination number of a non-complete graph G is the minimum cardinality among all independent semitotal dominating sets of G. In this article, we establish tight bounds and derive closed formulas for the independent semitotal domination number of lexicographic product graphs <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$G\circ H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>∘</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation>, expressed in terms of invariants of the factor graphs G and H.

On the Independent Semitotal Domination in Lexicographic Product Graphs

期刊论文
发表日期：

2025年11月29日

查看全文

A. Cabrera-Martínez,
J. L. López-Carmona,
A. Serrano-Díaz

摘要

Let G be a nontrivial connected graph and let $d_G(u,v)$ $d_{G} (u, v)$ be the distance between the vertices u and v in G. Let $\mathcal{I}\mathcal{D}(G)$ $I D (G)$ be the set of independent dominating sets of G. A set $I\in \mathcal{I}\mathcal{D}(G)$ $I \in I D (G)$ is an independent semitotal dominating set of G if for every vertex $v\in I$ $v \in I$ there exists a vertex $u\in I\setminus \{v\}$ $u \in I \ {v}$ such that $d_G(u,v)=2$ $d_{G} (u, v) = 2$ . The independent semitotal domination number of a non-complete graph G is the minimum cardinality among all independent semitotal dominating sets of G. In this article, we establish tight bounds and derive closed formulas for the independent semitotal domination number of lexicographic product graphs $G\circ H$ $G \circ H$ , expressed in terms of invariants of the factor graphs G and H.