Isolation Number of Transition Graphs

Isolation Number of Transition Graphs

Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo>...

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Bibliographic Details
Main Authors: Junhao Qu, Shumin Zhang
Format: Article
Language:English
Published: MDPI AG 2024-12-01
Series:Mathematics
Subjects:
partial domination
isolation number
bipartite graph
total graph
central graph
Online Access:https://www.mdpi.com/2227-7390/13/1/116
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Summary:Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></semantics></math></inline-formula> be a graph and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> be a family of graphs; a subset (<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></mrow></semantics></math></inline-formula>) is said to be an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>-isolating set if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>[</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>∖</mo><msub><mi>N</mi><mi>G</mi></msub><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow><mo>]</mo></mrow></semantics></math></inline-formula> does not contain <i>F</i> as a subgraph for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>∈</mo><mi mathvariant="script">F</mi></mrow></semantics></math></inline-formula>. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>-isolation number of <i>G</i> is the minimum cardinality of an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>-isolating set (<i>S</i>) of <i>G</i>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi mathvariant="script">F</mi><mo>)</mo></mrow></semantics></math></inline-formula>. When <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">F</mi><mo>=</mo><mo>{</mo><msub><mi>K</mi><mrow><mn>1</mn><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><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>ι</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> to define the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>-isolation number (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ι</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi mathvariant="script">F</mi><mo>)</mo></mrow></semantics></math></inline-formula>). In particular, when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, we use the short form of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ι</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> instead of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ι</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. A subset (<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></mrow></semantics></math></inline-formula>) is called an isolating set if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>∖</mo></mrow><msub><mi>N</mi><mi>G</mi></msub><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></mrow></semantics></math></inline-formula> is an independent set of <i>G</i>. The isolation number of <i>G</i> is the minimum cardinality of an isolating set, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ι</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>. In this paper, we mainly focus on research on the isolation number and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>-isolation number of a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> graph, total graph and central graph of graph <i>G</i>.
ISSN:2227-7390

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