A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs
Abstract The metric dimension dim(G) of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z(G) of a graph G is the minimum cardinality of a set S of black vertices (whereas vertice...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Eroh, Linda [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2017 |
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| Schlagwörter: |
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| Anmerkung: |
© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2017 |
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| Übergeordnetes Werk: |
Enthalten in: Acta mathematica sinica - Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, 1985, 33(2017), 6 vom: 24. Feb., Seite 731-747 |
|---|---|
| Übergeordnetes Werk: |
volume:33 ; year:2017 ; number:6 ; day:24 ; month:02 ; pages:731-747 |
| Links: |
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| DOI / URN: |
10.1007/s10114-017-4699-4 |
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OLC2062522673 |
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| 520 | |a Abstract The metric dimension dim(G) of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z(G) of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V(G)S are colored white) such that V(G) is turned black after finitely many applications of “the color-change rule”: a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim(T) ≤ Z(T) for a tree T, and that dim(G) ≤ Z(G)+1 if G is a unicyclic graph; along the way, we characterize trees T attaining dim(T) = Z(T). For a general graph G, we introduce the “cycle rank conjecture”. We conclude with a proof of dim(T) − 2 ≤ dim(T + e) ≤ dim(T) + 1 for $$e \in E\left( {\bar T} \right)$$. | ||
| 650 | 4 | |a Distance | |
| 650 | 4 | |a resolving set | |
| 650 | 4 | |a metric dimension | |
| 650 | 4 | |a zero forcing set | |
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| 650 | 4 | |a tree | |
| 650 | 4 | |a unicyclic graph | |
| 650 | 4 | |a cycle rank | |
| 700 | 1 | |a Kang, Cong X. |4 aut | |
| 700 | 1 | |a Yi, Eunjeong |4 aut | |
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10.1007/s10114-017-4699-4 doi (DE-627)OLC2062522673 (DE-He213)s10114-017-4699-4-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Eroh, Linda verfasserin aut A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2017 Abstract The metric dimension dim(G) of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z(G) of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V(G)S are colored white) such that V(G) is turned black after finitely many applications of “the color-change rule”: a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim(T) ≤ Z(T) for a tree T, and that dim(G) ≤ Z(G)+1 if G is a unicyclic graph; along the way, we characterize trees T attaining dim(T) = Z(T). For a general graph G, we introduce the “cycle rank conjecture”. We conclude with a proof of dim(T) − 2 ≤ dim(T + e) ≤ dim(T) + 1 for $$e \in E\left( {\bar T} \right)$$. Distance resolving set metric dimension zero forcing set zero forcing number tree unicyclic graph cycle rank Kang, Cong X. aut Yi, Eunjeong aut Enthalten in Acta mathematica sinica Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, 1985 33(2017), 6 vom: 24. Feb., Seite 731-747 (DE-627)129236772 (DE-600)58083-1 (DE-576)091206189 1000-9574 nnns volume:33 year:2017 number:6 day:24 month:02 pages:731-747 https://doi.org/10.1007/s10114-017-4699-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2018 AR 33 2017 6 24 02 731-747 |
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10.1007/s10114-017-4699-4 doi (DE-627)OLC2062522673 (DE-He213)s10114-017-4699-4-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Eroh, Linda verfasserin aut A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2017 Abstract The metric dimension dim(G) of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z(G) of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V(G)S are colored white) such that V(G) is turned black after finitely many applications of “the color-change rule”: a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim(T) ≤ Z(T) for a tree T, and that dim(G) ≤ Z(G)+1 if G is a unicyclic graph; along the way, we characterize trees T attaining dim(T) = Z(T). For a general graph G, we introduce the “cycle rank conjecture”. We conclude with a proof of dim(T) − 2 ≤ dim(T + e) ≤ dim(T) + 1 for $$e \in E\left( {\bar T} \right)$$. Distance resolving set metric dimension zero forcing set zero forcing number tree unicyclic graph cycle rank Kang, Cong X. aut Yi, Eunjeong aut Enthalten in Acta mathematica sinica Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, 1985 33(2017), 6 vom: 24. Feb., Seite 731-747 (DE-627)129236772 (DE-600)58083-1 (DE-576)091206189 1000-9574 nnns volume:33 year:2017 number:6 day:24 month:02 pages:731-747 https://doi.org/10.1007/s10114-017-4699-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2018 AR 33 2017 6 24 02 731-747 |
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10.1007/s10114-017-4699-4 doi (DE-627)OLC2062522673 (DE-He213)s10114-017-4699-4-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Eroh, Linda verfasserin aut A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2017 Abstract The metric dimension dim(G) of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z(G) of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V(G)S are colored white) such that V(G) is turned black after finitely many applications of “the color-change rule”: a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim(T) ≤ Z(T) for a tree T, and that dim(G) ≤ Z(G)+1 if G is a unicyclic graph; along the way, we characterize trees T attaining dim(T) = Z(T). For a general graph G, we introduce the “cycle rank conjecture”. We conclude with a proof of dim(T) − 2 ≤ dim(T + e) ≤ dim(T) + 1 for $$e \in E\left( {\bar T} \right)$$. Distance resolving set metric dimension zero forcing set zero forcing number tree unicyclic graph cycle rank Kang, Cong X. aut Yi, Eunjeong aut Enthalten in Acta mathematica sinica Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, 1985 33(2017), 6 vom: 24. Feb., Seite 731-747 (DE-627)129236772 (DE-600)58083-1 (DE-576)091206189 1000-9574 nnns volume:33 year:2017 number:6 day:24 month:02 pages:731-747 https://doi.org/10.1007/s10114-017-4699-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2018 AR 33 2017 6 24 02 731-747 |
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10.1007/s10114-017-4699-4 doi (DE-627)OLC2062522673 (DE-He213)s10114-017-4699-4-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Eroh, Linda verfasserin aut A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2017 Abstract The metric dimension dim(G) of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z(G) of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V(G)S are colored white) such that V(G) is turned black after finitely many applications of “the color-change rule”: a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim(T) ≤ Z(T) for a tree T, and that dim(G) ≤ Z(G)+1 if G is a unicyclic graph; along the way, we characterize trees T attaining dim(T) = Z(T). For a general graph G, we introduce the “cycle rank conjecture”. We conclude with a proof of dim(T) − 2 ≤ dim(T + e) ≤ dim(T) + 1 for $$e \in E\left( {\bar T} \right)$$. Distance resolving set metric dimension zero forcing set zero forcing number tree unicyclic graph cycle rank Kang, Cong X. aut Yi, Eunjeong aut Enthalten in Acta mathematica sinica Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, 1985 33(2017), 6 vom: 24. Feb., Seite 731-747 (DE-627)129236772 (DE-600)58083-1 (DE-576)091206189 1000-9574 nnns volume:33 year:2017 number:6 day:24 month:02 pages:731-747 https://doi.org/10.1007/s10114-017-4699-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2018 AR 33 2017 6 24 02 731-747 |
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10.1007/s10114-017-4699-4 doi (DE-627)OLC2062522673 (DE-He213)s10114-017-4699-4-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Eroh, Linda verfasserin aut A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2017 Abstract The metric dimension dim(G) of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z(G) of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V(G)S are colored white) such that V(G) is turned black after finitely many applications of “the color-change rule”: a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim(T) ≤ Z(T) for a tree T, and that dim(G) ≤ Z(G)+1 if G is a unicyclic graph; along the way, we characterize trees T attaining dim(T) = Z(T). For a general graph G, we introduce the “cycle rank conjecture”. We conclude with a proof of dim(T) − 2 ≤ dim(T + e) ≤ dim(T) + 1 for $$e \in E\left( {\bar T} \right)$$. Distance resolving set metric dimension zero forcing set zero forcing number tree unicyclic graph cycle rank Kang, Cong X. aut Yi, Eunjeong aut Enthalten in Acta mathematica sinica Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, 1985 33(2017), 6 vom: 24. Feb., Seite 731-747 (DE-627)129236772 (DE-600)58083-1 (DE-576)091206189 1000-9574 nnns volume:33 year:2017 number:6 day:24 month:02 pages:731-747 https://doi.org/10.1007/s10114-017-4699-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2018 AR 33 2017 6 24 02 731-747 |
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a comparison between the metric dimension and zero forcing number of trees and unicyclic graphs |
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A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs |
| abstract |
Abstract The metric dimension dim(G) of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z(G) of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V(G)S are colored white) such that V(G) is turned black after finitely many applications of “the color-change rule”: a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim(T) ≤ Z(T) for a tree T, and that dim(G) ≤ Z(G)+1 if G is a unicyclic graph; along the way, we characterize trees T attaining dim(T) = Z(T). For a general graph G, we introduce the “cycle rank conjecture”. We conclude with a proof of dim(T) − 2 ≤ dim(T + e) ≤ dim(T) + 1 for $$e \in E\left( {\bar T} \right)$$. © Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2017 |
| abstractGer |
Abstract The metric dimension dim(G) of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z(G) of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V(G)S are colored white) such that V(G) is turned black after finitely many applications of “the color-change rule”: a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim(T) ≤ Z(T) for a tree T, and that dim(G) ≤ Z(G)+1 if G is a unicyclic graph; along the way, we characterize trees T attaining dim(T) = Z(T). For a general graph G, we introduce the “cycle rank conjecture”. We conclude with a proof of dim(T) − 2 ≤ dim(T + e) ≤ dim(T) + 1 for $$e \in E\left( {\bar T} \right)$$. © Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2017 |
| abstract_unstemmed |
Abstract The metric dimension dim(G) of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z(G) of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V(G)S are colored white) such that V(G) is turned black after finitely many applications of “the color-change rule”: a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim(T) ≤ Z(T) for a tree T, and that dim(G) ≤ Z(G)+1 if G is a unicyclic graph; along the way, we characterize trees T attaining dim(T) = Z(T). For a general graph G, we introduce the “cycle rank conjecture”. We conclude with a proof of dim(T) − 2 ≤ dim(T + e) ≤ dim(T) + 1 for $$e \in E\left( {\bar T} \right)$$. © Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2017 |
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6 |
| title_short |
A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs |
| url |
https://doi.org/10.1007/s10114-017-4699-4 |
| remote_bool |
false |
| author2 |
Kang, Cong X. Yi, Eunjeong |
| author2Str |
Kang, Cong X. Yi, Eunjeong |
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| doi_str |
10.1007/s10114-017-4699-4 |
| up_date |
2024-07-03T15:22:53.069Z |
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1803571873002291200 |
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