Fuzzy Coalition Graphs: A Framework for Understanding Cooperative Dominance in Uncertain Networks
In a fuzzy graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">G</mi></semantics></math></inline-formula>, a fuzzy coalition is formed by two d...
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| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-11-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/12/22/3614 |
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| Summary: | In a fuzzy graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">G</mi></semantics></math></inline-formula>, a fuzzy coalition is formed by two disjoint vertex sets <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">V</mi><mn mathvariant="monospace">1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">V</mi><mn mathvariant="monospace">2</mn></msub></semantics></math></inline-formula>, neither of which is a strongly dominating set, but the union <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">V</mi><mn mathvariant="monospace">1</mn></msub><mo>∪</mo><msub><mi mathvariant="double-struck">V</mi><mn mathvariant="monospace">2</mn></msub></mrow></semantics></math></inline-formula> forms a strongly dominating set. A fuzzy coalition partition of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">G</mi></semantics></math></inline-formula> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Π</mo><mo>=</mo><mo>{</mo><msub><mi mathvariant="double-struck">V</mi><mn mathvariant="monospace">1</mn></msub><mo>,</mo><msub><mi mathvariant="double-struck">V</mi><mn mathvariant="monospace">2</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi mathvariant="double-struck">V</mi><mi mathvariant="double-struck">k</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula>, where each set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">V</mi><mi mathvariant="double-struck">i</mi></msub></semantics></math></inline-formula> either forms a singleton strongly dominating set or is not a strongly dominating set but forms a fuzzy coalition with another non-strongly dominating set in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Π</mo></semantics></math></inline-formula>. A fuzzy graph with such a fuzzy coalition partition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Π</mo></semantics></math></inline-formula> is called a fuzzy coalition graph, denoted as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">FG</mi><mo>(</mo><mi mathvariant="double-struck">G</mi><mo>,</mo><mo>Π</mo><mo>)</mo></mrow></semantics></math></inline-formula>. The vertex set of the fuzzy coalition graph consists of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi mathvariant="double-struck">V</mi><mn mathvariant="monospace">1</mn></msub><mo>,</mo><msub><mi mathvariant="double-struck">V</mi><mn mathvariant="monospace">2</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi mathvariant="double-struck">V</mi><mi mathvariant="double-struck">k</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula>, corresponding one-to-one with the sets of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Π</mo></semantics></math></inline-formula>, and the two vertices are adjacent in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">FG</mi><mo>(</mo><mi mathvariant="double-struck">G</mi><mo>,</mo><mo>Π</mo><mo>)</mo></mrow></semantics></math></inline-formula> if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">V</mi><mi mathvariant="double-struck">i</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">V</mi><mi mathvariant="double-struck">j</mi></msub></semantics></math></inline-formula> are fuzzy coalition partners in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Π</mo></semantics></math></inline-formula>. This study demonstrates how fuzzy coalition graphs can model and optimize cybersecurity collaborations across critical infrastructures in smart cities, ensuring comprehensive protection against cyber threats. This study concludes that fuzzy coalition graphs offer a robust framework for optimizing collaboration and decision-making in interconnected systems like smart cities. |
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| ISSN: | 2227-7390 |