On Non-Zero Vertex Signed Domination
For a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi&g...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-03-01
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| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/15/3/741 |
| _version_ | 1797608876066996224 |
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| author | Baogen Xu Mengmeng Zheng Ting Lan |
| author_facet | Baogen Xu Mengmeng Zheng Ting Lan |
| author_sort | Baogen Xu |
| collection | DOAJ |
| description | For a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>→</mo><mrow><mo>{</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mo>+</mo><mn>1</mn></mrow><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi></mrow></semantics></math></inline-formula> then we write <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>v</mi><mo>∈</mo><mi>S</mi></mrow></munder><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow></semantics></math></inline-formula>. A function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>f</mi></semantics></math></inline-formula> is said to be a non-zero vertex signed dominating function (for short, NVSDF) of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>N</mi><mrow><mo>[</mo><mi>v</mi><mo>]</mo></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> holds for every vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>v</mi></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula>, and the non-zero vertex signed domination number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>s</mi><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>max</mi><mrow><mo>{</mo><mrow><mi>f</mi><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mrow><mo>|</mo><mrow><mrow><mi>f</mi><mo> </mo><mi>is</mi><mo> </mo><mi>an</mi><mo> </mo><mi>NVSDF</mi><mo> </mo><mi>of</mi><mo> </mo></mrow><mi>G</mi></mrow></mrow></mrow><mo>}</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> In this paper, the novel concept of the non-zero vertex signed domination for graphs is introduced. There is also a special symmetry concept in graphs. Some upper bounds of the non-zero vertex signed domination number of a graph are given. The exact value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>s</mi><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for several special classes of graphs is determined. Finally, we pose some open problems. |
| first_indexed | 2024-03-11T05:50:40Z |
| format | Article |
| id | doaj.art-de9660ffb22a438b87f2843e315de0fe |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T05:50:40Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-de9660ffb22a438b87f2843e315de0fe2023-11-17T14:10:20ZengMDPI AGSymmetry2073-89942023-03-0115374110.3390/sym15030741On Non-Zero Vertex Signed DominationBaogen Xu0Mengmeng Zheng1Ting Lan2School of Mathematics, East China Jiaotong University, Nanchang 330013, ChinaSchool of Mathematics, East China Jiaotong University, Nanchang 330013, ChinaSchool of Mathematics, East China Jiaotong University, Nanchang 330013, ChinaFor a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>→</mo><mrow><mo>{</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mo>+</mo><mn>1</mn></mrow><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi></mrow></semantics></math></inline-formula> then we write <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>v</mi><mo>∈</mo><mi>S</mi></mrow></munder><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow></semantics></math></inline-formula>. A function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>f</mi></semantics></math></inline-formula> is said to be a non-zero vertex signed dominating function (for short, NVSDF) of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>N</mi><mrow><mo>[</mo><mi>v</mi><mo>]</mo></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> holds for every vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>v</mi></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula>, and the non-zero vertex signed domination number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>s</mi><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>max</mi><mrow><mo>{</mo><mrow><mi>f</mi><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mrow><mo>|</mo><mrow><mrow><mi>f</mi><mo> </mo><mi>is</mi><mo> </mo><mi>an</mi><mo> </mo><mi>NVSDF</mi><mo> </mo><mi>of</mi><mo> </mo></mrow><mi>G</mi></mrow></mrow></mrow><mo>}</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> In this paper, the novel concept of the non-zero vertex signed domination for graphs is introduced. There is also a special symmetry concept in graphs. Some upper bounds of the non-zero vertex signed domination number of a graph are given. The exact value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>s</mi><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for several special classes of graphs is determined. Finally, we pose some open problems.https://www.mdpi.com/2073-8994/15/3/741graphnon-zero vertex signed dominationnon-zero vertex signed domination number |
| spellingShingle | Baogen Xu Mengmeng Zheng Ting Lan On Non-Zero Vertex Signed Domination Symmetry graph non-zero vertex signed domination non-zero vertex signed domination number |
| title | On Non-Zero Vertex Signed Domination |
| title_full | On Non-Zero Vertex Signed Domination |
| title_fullStr | On Non-Zero Vertex Signed Domination |
| title_full_unstemmed | On Non-Zero Vertex Signed Domination |
| title_short | On Non-Zero Vertex Signed Domination |
| title_sort | on non zero vertex signed domination |
| topic | graph non-zero vertex signed domination non-zero vertex signed domination number |
| url | https://www.mdpi.com/2073-8994/15/3/741 |
| work_keys_str_mv | AT baogenxu onnonzerovertexsigneddomination AT mengmengzheng onnonzerovertexsigneddomination AT tinglan onnonzerovertexsigneddomination |