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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2024-12-01
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| author | Jiatong Cui Tianhao Li Jiayuan Zhang Xiaodong Chen Liming Xiong |
| author_facet | Jiatong Cui Tianhao Li Jiayuan Zhang Xiaodong Chen Liming Xiong |
| author_sort | Jiatong Cui |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-89778b7225cc420ca6aa7536507b009a |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-89778b7225cc420ca6aa7536507b009a2025-01-24T13:22:08ZengMDPI AGAxioms2075-16802024-12-011411010.3390/axioms14010010Co-Secure Domination Number in Some GraphsJiatong Cui0Tianhao Li1Jiayuan Zhang2Xiaodong Chen3Liming Xiong4School of Mathematics, Liaoning Normal University, Dalian 116029, ChinaSchool of Mathematics, Liaoning Normal University, Dalian 116029, ChinaSchool of Mathematics, Liaoning Normal University, Dalian 116029, ChinaSchool of Mathematics, Liaoning Normal University, Dalian 116029, ChinaSchool of Mathematics and Statistics, Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, ChinaLet <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.https://www.mdpi.com/2075-1680/14/1/10co-secure dominating setsecure dominating setco-secure domination number |
| spellingShingle | Jiatong Cui Tianhao Li Jiayuan Zhang Xiaodong Chen Liming Xiong Co-Secure Domination Number in Some Graphs Axioms co-secure dominating set secure dominating set co-secure domination number |
| title | Co-Secure Domination Number in Some Graphs |
| title_full | Co-Secure Domination Number in Some Graphs |
| title_fullStr | Co-Secure Domination Number in Some Graphs |
| title_full_unstemmed | Co-Secure Domination Number in Some Graphs |
| title_short | Co-Secure Domination Number in Some Graphs |
| title_sort | co secure domination number in some graphs |
| topic | co-secure dominating set secure dominating set co-secure domination number |
| url | https://www.mdpi.com/2075-1680/14/1/10 |
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