Co-Secure Domination Number in Some Graphs

Co-Secure Domination Number in Some Graphs

Let <i>G</i> be a graph and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo&...

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Bibliographic Details
Main Authors: Jiatong Cui, Tianhao Li, Jiayuan Zhang, Xiaodong Chen, Liming Xiong
Format: Article
Language:English
Published: MDPI AG 2024-12-01
Series:Axioms
Subjects:
co-secure dominating set
secure dominating set
co-secure domination number
Online Access:https://www.mdpi.com/2075-1680/14/1/10
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Summary:Let <i>G</i> be a graph and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> If <i>S</i> is a dominating set of <i>G</i>, and for each vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mi>S</mi><mo>,</mo></mrow></semantics></math></inline-formula> there is a neighbor of <i>u</i> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>,</mo></mrow></semantics></math></inline-formula> denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>,</mo></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>S</mi><mo>∖</mo><mo>{</mo><mi>v</mi><mo>}</mo><mo>)</mo><mo>∪</mo><mo>{</mo><mi>u</mi><mo>}</mo></mrow></semantics></math></inline-formula> is a dominating set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>,</mo></mrow></semantics></math></inline-formula> then <i>S</i> is a secure dominating set (SDS) of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>.</mo></mrow></semantics></math></inline-formula> Conversely, <i>S</i> is a co-secure dominating set (CSDS) of <i>G</i> if <i>S</i> is a dominating set of <i>G</i> and for each vertex <i>v</i> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>,</mo></mrow></semantics></math></inline-formula> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mi>S</mi></mrow></semantics></math></inline-formula> contains a neighbor of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>,</mo></mrow></semantics></math></inline-formula> denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>,</mo></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>S</mi><mo>∖</mo><mo>{</mo><mi>v</mi><mo>}</mo><mo>)</mo><mo>∪</mo><mo>{</mo><mi>u</mi><mo>}</mo></mrow></semantics></math></inline-formula> is a dominating set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>.</mo></mrow></semantics></math></inline-formula> The minimum cardinality of a CSDS (resp. SDS) of <i>G</i> is the co-secure (resp. secure) domination number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>.</mo></mrow></semantics></math></inline-formula> We use <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> to denote the co-secure domination number and secure domination number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>,</mo></mrow></semantics></math></inline-formula> respectively. Arumugam et al. proposed two questions: (1) Characterize a graph <i>G</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the independence number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>;</mo></mrow></semantics></math></inline-formula> (2) Characterize a graph <i>G</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>γ</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> In this paper, we characterize some forbidden induced subgraphs for a graph <i>G</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>; moreover, we obtain that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>γ</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for each <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>K</mi><mn>3</mn></msub><mo>,</mo><msub><mi>C</mi><mn>5</mn></msub><mo>,</mo><msub><mi>P</mi><mn>5</mn></msub><mo>}</mo></mrow></semantics></math></inline-formula>-free graph <i>G</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>2</mn><mo>.</mo></mrow></semantics></math></inline-formula> Our conclusions can generalize some known results.
ISSN:2075-1680

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