On Two Outer Independent Roman Domination Related Parameters in Torus Graphs
In a graph <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>)<...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/18/3361 |
| _version_ | 1797485190383140864 |
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| author | Hong Gao Xing Liu Yuanyuan Guo Yuansheng Yang |
| author_facet | Hong Gao Xing Liu Yuanyuan Guo Yuansheng Yang |
| author_sort | Hong Gao |
| collection | DOAJ |
| description | In a graph <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>, where every vertex is assigned 0, 1 or 2, <i>f</i> is an assignment such that every vertex assigned 0 has at least one neighbor assigned 2 and all vertices labeled by 0 are independent, then <i>f</i> is called an outer independent Roman dominating function (OIRDF). The domination is strengthened if every vertex is assigned 0, 1, 2 or 3, <i>f</i> is such an assignment that each vertex assigned 0 has at least two neighbors assigned 2 or one neighbor assigned 3, each vertex assigned 1 has at least one neighbor assigned 2 or 3, and all vertices labeled by 0 are independent, then <i>f</i> is called an outer independent double Roman dominating function (OIDRDF). The weight of an (OIDRDF) OIRDF <i>f</i> is the sum of <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></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></semantics></math></inline-formula>. The outer independent (double) Roman domination number (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>o</mi><mi>i</mi><mi>d</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>o</mi><mi>i</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the minimum weight taken over all (OIDRDFs) OIRDFs of <i>G</i>. In this article, we investigate these two parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>o</mi><mi>i</mi><mi>R</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><mrow><mi>o</mi><mi>i</mi><mi>d</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of regular graphs and present lower bounds on them. We improve the lower bound on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>o</mi><mi>i</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for a regular graph presented by Ahangar et al. (2017). Furthermore, we present upper bounds on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>o</mi><mi>i</mi><mi>R</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><mrow><mi>o</mi><mi>i</mi><mi>d</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for torus graphs. Furthermore, we determine the exact values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>o</mi><mi>i</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><msub><mi>C</mi><mn>3</mn></msub><mo>□</mo><msub><mi>C</mi><mi>n</mi></msub><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><mrow><mi>o</mi><mi>i</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><msub><mi>C</mi><mi>m</mi></msub><mo>□</mo><msub><mi>C</mi><mi>n</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≡</mo><mn>0</mn><mspace width="4.44443pt"></mspace><mo>(</mo><mi>mod</mi><mspace width="0.277778em"></mspace><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≡</mo><mn>0</mn><mspace width="4.44443pt"></mspace><mo>(</mo><mi>mod</mi><mspace width="0.277778em"></mspace><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and 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>o</mi><mi>i</mi><mi>d</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><msub><mi>C</mi><mn>3</mn></msub><mo>□</mo><msub><mi>C</mi><mi>n</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. By our result, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>o</mi><mi>i</mi><mi>d</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><msub><mi>C</mi><mi>m</mi></msub><mo>□</mo><msub><mi>C</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>≤</mo><mn>5</mn><mi>m</mi><mi>n</mi><mo>/</mo><mn>4</mn></mrow></semantics></math></inline-formula> which verifies the open question is correct for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>m</mi></msub><mo>□</mo><msub><mi>C</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> that was presented by Ahangar et al. (2020). |
| first_indexed | 2024-03-09T23:15:14Z |
| format | Article |
| id | doaj.art-fb222e2809764ec58a00d94e2d96f513 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T23:15:14Z |
| publishDate | 2022-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-fb222e2809764ec58a00d94e2d96f5132023-11-23T17:37:28ZengMDPI AGMathematics2227-73902022-09-011018336110.3390/math10183361On Two Outer Independent Roman Domination Related Parameters in Torus GraphsHong Gao0Xing Liu1Yuanyuan Guo2Yuansheng Yang3College of Science, Dalian Maritime University, Dalian 116026, ChinaCollege of Science, Dalian Maritime University, Dalian 116026, ChinaCollege of Science, Dalian Maritime University, Dalian 116026, ChinaSchool of Computer Science and Technology, Dalian University of Technology, Dalian 116024, ChinaIn a graph <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>, where every vertex is assigned 0, 1 or 2, <i>f</i> is an assignment such that every vertex assigned 0 has at least one neighbor assigned 2 and all vertices labeled by 0 are independent, then <i>f</i> is called an outer independent Roman dominating function (OIRDF). The domination is strengthened if every vertex is assigned 0, 1, 2 or 3, <i>f</i> is such an assignment that each vertex assigned 0 has at least two neighbors assigned 2 or one neighbor assigned 3, each vertex assigned 1 has at least one neighbor assigned 2 or 3, and all vertices labeled by 0 are independent, then <i>f</i> is called an outer independent double Roman dominating function (OIDRDF). The weight of an (OIDRDF) OIRDF <i>f</i> is the sum of <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></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></semantics></math></inline-formula>. The outer independent (double) Roman domination number (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>o</mi><mi>i</mi><mi>d</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>o</mi><mi>i</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the minimum weight taken over all (OIDRDFs) OIRDFs of <i>G</i>. In this article, we investigate these two parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>o</mi><mi>i</mi><mi>R</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><mrow><mi>o</mi><mi>i</mi><mi>d</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of regular graphs and present lower bounds on them. We improve the lower bound on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>o</mi><mi>i</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for a regular graph presented by Ahangar et al. (2017). Furthermore, we present upper bounds on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>o</mi><mi>i</mi><mi>R</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><mrow><mi>o</mi><mi>i</mi><mi>d</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for torus graphs. Furthermore, we determine the exact values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>o</mi><mi>i</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><msub><mi>C</mi><mn>3</mn></msub><mo>□</mo><msub><mi>C</mi><mi>n</mi></msub><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><mrow><mi>o</mi><mi>i</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><msub><mi>C</mi><mi>m</mi></msub><mo>□</mo><msub><mi>C</mi><mi>n</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≡</mo><mn>0</mn><mspace width="4.44443pt"></mspace><mo>(</mo><mi>mod</mi><mspace width="0.277778em"></mspace><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≡</mo><mn>0</mn><mspace width="4.44443pt"></mspace><mo>(</mo><mi>mod</mi><mspace width="0.277778em"></mspace><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and 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>o</mi><mi>i</mi><mi>d</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><msub><mi>C</mi><mn>3</mn></msub><mo>□</mo><msub><mi>C</mi><mi>n</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. By our result, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>o</mi><mi>i</mi><mi>d</mi><mi>R</mi></mrow></msub><mrow><mo>(</mo><msub><mi>C</mi><mi>m</mi></msub><mo>□</mo><msub><mi>C</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>≤</mo><mn>5</mn><mi>m</mi><mi>n</mi><mo>/</mo><mn>4</mn></mrow></semantics></math></inline-formula> which verifies the open question is correct for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>m</mi></msub><mo>□</mo><msub><mi>C</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> that was presented by Ahangar et al. (2020).https://www.mdpi.com/2227-7390/10/18/3361regular graphsouter independent double Roman dominationCartesian product of cyclesouter independent Roman domination |
| spellingShingle | Hong Gao Xing Liu Yuanyuan Guo Yuansheng Yang On Two Outer Independent Roman Domination Related Parameters in Torus Graphs Mathematics regular graphs outer independent double Roman domination Cartesian product of cycles outer independent Roman domination |
| title | On Two Outer Independent Roman Domination Related Parameters in Torus Graphs |
| title_full | On Two Outer Independent Roman Domination Related Parameters in Torus Graphs |
| title_fullStr | On Two Outer Independent Roman Domination Related Parameters in Torus Graphs |
| title_full_unstemmed | On Two Outer Independent Roman Domination Related Parameters in Torus Graphs |
| title_short | On Two Outer Independent Roman Domination Related Parameters in Torus Graphs |
| title_sort | on two outer independent roman domination related parameters in torus graphs |
| topic | regular graphs outer independent double Roman domination Cartesian product of cycles outer independent Roman domination |
| url | https://www.mdpi.com/2227-7390/10/18/3361 |
| work_keys_str_mv | AT honggao ontwoouterindependentromandominationrelatedparametersintorusgraphs AT xingliu ontwoouterindependentromandominationrelatedparametersintorusgraphs AT yuanyuanguo ontwoouterindependentromandominationrelatedparametersintorusgraphs AT yuanshengyang ontwoouterindependentromandominationrelatedparametersintorusgraphs |