On metric dimension of permutation 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 set of chosen vertices. Let $$G_1$$ and $$G_2$$ be disjoint copies of a graph $$G$$, and let $$\sigma : V(G_1) \righta...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Hallaway, Michael [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2013 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media New York 2012 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of combinatorial optimization - Springer US, 1997, 28(2013), 4 vom: 03. Jan., Seite 814-826 |
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| Übergeordnetes Werk: |
volume:28 ; year:2013 ; number:4 ; day:03 ; month:01 ; pages:814-826 |
| Links: |
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| DOI / URN: |
10.1007/s10878-012-9587-3 |
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| Katalog-ID: |
OLC2044617919 |
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| 520 | |a Abstract The metric dimension$$\dim (G)$$ of a graph $$G$$ is the minimum number of vertices such that every vertex of $$G$$ is uniquely determined by its vector of distances to the set of chosen vertices. Let $$G_1$$ and $$G_2$$ be disjoint copies of a graph $$G$$, and let $$\sigma : V(G_1) \rightarrow V(G_2)$$ be a permutation. Then, a permutation graph$$G_{\sigma }=(V, E)$$ has the vertex set $$V=V(G_1) \cup V(G_2)$$ and the edge set $$E=E(G_1) \cup E(G_2) \cup \{uv \mid v=\sigma (u)\}$$. We show that $$2 \le \dim (G_{\sigma }) \le n-1$$ for any connected graph $$G$$ of order $$n$$ at least $$3$$. We give examples showing that neither is there a function $$f$$ such that $$\dim (G)<f(\dim (G_{\sigma }))$$ for all pairs $$(G,\sigma )$$, nor is there a function $$g$$ such that $$g(\dim (G))>\dim (G_{\sigma })$$ for all pairs $$(G, \sigma )$$. Further, we characterize permutation graphs $$G_{\sigma }$$ satisfying $$\dim (G_{\sigma })=n-1$$ when $$G$$ is a complete $$k$$-partite graph, a cycle, or a path on $$n$$ vertices. | ||
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| 650 | 4 | |a Permutation graph | |
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| 650 | 4 | |a Path | |
| 700 | 1 | |a Kang, Cong X. |4 aut | |
| 700 | 1 | |a Yi, Eunjeong |4 aut | |
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10.1007/s10878-012-9587-3 doi (DE-627)OLC2044617919 (DE-He213)s10878-012-9587-3-p DE-627 ger DE-627 rakwb eng 510 VZ 3,2 ssgn Hallaway, Michael verfasserin aut On metric dimension of permutation graphs 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2012 Abstract The metric dimension$$\dim (G)$$ of a graph $$G$$ is the minimum number of vertices such that every vertex of $$G$$ is uniquely determined by its vector of distances to the set of chosen vertices. Let $$G_1$$ and $$G_2$$ be disjoint copies of a graph $$G$$, and let $$\sigma : V(G_1) \rightarrow V(G_2)$$ be a permutation. Then, a permutation graph$$G_{\sigma }=(V, E)$$ has the vertex set $$V=V(G_1) \cup V(G_2)$$ and the edge set $$E=E(G_1) \cup E(G_2) \cup \{uv \mid v=\sigma (u)\}$$. We show that $$2 \le \dim (G_{\sigma }) \le n-1$$ for any connected graph $$G$$ of order $$n$$ at least $$3$$. We give examples showing that neither is there a function $$f$$ such that $$\dim (G)<f(\dim (G_{\sigma }))$$ for all pairs $$(G,\sigma )$$, nor is there a function $$g$$ such that $$g(\dim (G))>\dim (G_{\sigma })$$ for all pairs $$(G, \sigma )$$. Further, we characterize permutation graphs $$G_{\sigma }$$ satisfying $$\dim (G_{\sigma })=n-1$$ when $$G$$ is a complete $$k$$-partite graph, a cycle, or a path on $$n$$ vertices. Metric dimension Permutation graph Complete -partite graph Cycle Path Kang, Cong X. aut Yi, Eunjeong aut Enthalten in Journal of combinatorial optimization Springer US, 1997 28(2013), 4 vom: 03. Jan., Seite 814-826 (DE-627)216539323 (DE-600)1339574-9 (DE-576)094421935 1382-6905 nnns volume:28 year:2013 number:4 day:03 month:01 pages:814-826 https://doi.org/10.1007/s10878-012-9587-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_24 GBV_ILN_26 GBV_ILN_70 GBV_ILN_2108 AR 28 2013 4 03 01 814-826 |
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10.1007/s10878-012-9587-3 doi (DE-627)OLC2044617919 (DE-He213)s10878-012-9587-3-p DE-627 ger DE-627 rakwb eng 510 VZ 3,2 ssgn Hallaway, Michael verfasserin aut On metric dimension of permutation graphs 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2012 Abstract The metric dimension$$\dim (G)$$ of a graph $$G$$ is the minimum number of vertices such that every vertex of $$G$$ is uniquely determined by its vector of distances to the set of chosen vertices. Let $$G_1$$ and $$G_2$$ be disjoint copies of a graph $$G$$, and let $$\sigma : V(G_1) \rightarrow V(G_2)$$ be a permutation. Then, a permutation graph$$G_{\sigma }=(V, E)$$ has the vertex set $$V=V(G_1) \cup V(G_2)$$ and the edge set $$E=E(G_1) \cup E(G_2) \cup \{uv \mid v=\sigma (u)\}$$. We show that $$2 \le \dim (G_{\sigma }) \le n-1$$ for any connected graph $$G$$ of order $$n$$ at least $$3$$. We give examples showing that neither is there a function $$f$$ such that $$\dim (G)<f(\dim (G_{\sigma }))$$ for all pairs $$(G,\sigma )$$, nor is there a function $$g$$ such that $$g(\dim (G))>\dim (G_{\sigma })$$ for all pairs $$(G, \sigma )$$. Further, we characterize permutation graphs $$G_{\sigma }$$ satisfying $$\dim (G_{\sigma })=n-1$$ when $$G$$ is a complete $$k$$-partite graph, a cycle, or a path on $$n$$ vertices. Metric dimension Permutation graph Complete -partite graph Cycle Path Kang, Cong X. aut Yi, Eunjeong aut Enthalten in Journal of combinatorial optimization Springer US, 1997 28(2013), 4 vom: 03. Jan., Seite 814-826 (DE-627)216539323 (DE-600)1339574-9 (DE-576)094421935 1382-6905 nnns volume:28 year:2013 number:4 day:03 month:01 pages:814-826 https://doi.org/10.1007/s10878-012-9587-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_24 GBV_ILN_26 GBV_ILN_70 GBV_ILN_2108 AR 28 2013 4 03 01 814-826 |
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10.1007/s10878-012-9587-3 doi (DE-627)OLC2044617919 (DE-He213)s10878-012-9587-3-p DE-627 ger DE-627 rakwb eng 510 VZ 3,2 ssgn Hallaway, Michael verfasserin aut On metric dimension of permutation graphs 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2012 Abstract The metric dimension$$\dim (G)$$ of a graph $$G$$ is the minimum number of vertices such that every vertex of $$G$$ is uniquely determined by its vector of distances to the set of chosen vertices. Let $$G_1$$ and $$G_2$$ be disjoint copies of a graph $$G$$, and let $$\sigma : V(G_1) \rightarrow V(G_2)$$ be a permutation. Then, a permutation graph$$G_{\sigma }=(V, E)$$ has the vertex set $$V=V(G_1) \cup V(G_2)$$ and the edge set $$E=E(G_1) \cup E(G_2) \cup \{uv \mid v=\sigma (u)\}$$. We show that $$2 \le \dim (G_{\sigma }) \le n-1$$ for any connected graph $$G$$ of order $$n$$ at least $$3$$. We give examples showing that neither is there a function $$f$$ such that $$\dim (G)<f(\dim (G_{\sigma }))$$ for all pairs $$(G,\sigma )$$, nor is there a function $$g$$ such that $$g(\dim (G))>\dim (G_{\sigma })$$ for all pairs $$(G, \sigma )$$. Further, we characterize permutation graphs $$G_{\sigma }$$ satisfying $$\dim (G_{\sigma })=n-1$$ when $$G$$ is a complete $$k$$-partite graph, a cycle, or a path on $$n$$ vertices. Metric dimension Permutation graph Complete -partite graph Cycle Path Kang, Cong X. aut Yi, Eunjeong aut Enthalten in Journal of combinatorial optimization Springer US, 1997 28(2013), 4 vom: 03. Jan., Seite 814-826 (DE-627)216539323 (DE-600)1339574-9 (DE-576)094421935 1382-6905 nnns volume:28 year:2013 number:4 day:03 month:01 pages:814-826 https://doi.org/10.1007/s10878-012-9587-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_24 GBV_ILN_26 GBV_ILN_70 GBV_ILN_2108 AR 28 2013 4 03 01 814-826 |
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10.1007/s10878-012-9587-3 doi (DE-627)OLC2044617919 (DE-He213)s10878-012-9587-3-p DE-627 ger DE-627 rakwb eng 510 VZ 3,2 ssgn Hallaway, Michael verfasserin aut On metric dimension of permutation graphs 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2012 Abstract The metric dimension$$\dim (G)$$ of a graph $$G$$ is the minimum number of vertices such that every vertex of $$G$$ is uniquely determined by its vector of distances to the set of chosen vertices. Let $$G_1$$ and $$G_2$$ be disjoint copies of a graph $$G$$, and let $$\sigma : V(G_1) \rightarrow V(G_2)$$ be a permutation. Then, a permutation graph$$G_{\sigma }=(V, E)$$ has the vertex set $$V=V(G_1) \cup V(G_2)$$ and the edge set $$E=E(G_1) \cup E(G_2) \cup \{uv \mid v=\sigma (u)\}$$. We show that $$2 \le \dim (G_{\sigma }) \le n-1$$ for any connected graph $$G$$ of order $$n$$ at least $$3$$. We give examples showing that neither is there a function $$f$$ such that $$\dim (G)<f(\dim (G_{\sigma }))$$ for all pairs $$(G,\sigma )$$, nor is there a function $$g$$ such that $$g(\dim (G))>\dim (G_{\sigma })$$ for all pairs $$(G, \sigma )$$. Further, we characterize permutation graphs $$G_{\sigma }$$ satisfying $$\dim (G_{\sigma })=n-1$$ when $$G$$ is a complete $$k$$-partite graph, a cycle, or a path on $$n$$ vertices. Metric dimension Permutation graph Complete -partite graph Cycle Path Kang, Cong X. aut Yi, Eunjeong aut Enthalten in Journal of combinatorial optimization Springer US, 1997 28(2013), 4 vom: 03. Jan., Seite 814-826 (DE-627)216539323 (DE-600)1339574-9 (DE-576)094421935 1382-6905 nnns volume:28 year:2013 number:4 day:03 month:01 pages:814-826 https://doi.org/10.1007/s10878-012-9587-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_24 GBV_ILN_26 GBV_ILN_70 GBV_ILN_2108 AR 28 2013 4 03 01 814-826 |
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10.1007/s10878-012-9587-3 doi (DE-627)OLC2044617919 (DE-He213)s10878-012-9587-3-p DE-627 ger DE-627 rakwb eng 510 VZ 3,2 ssgn Hallaway, Michael verfasserin aut On metric dimension of permutation graphs 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2012 Abstract The metric dimension$$\dim (G)$$ of a graph $$G$$ is the minimum number of vertices such that every vertex of $$G$$ is uniquely determined by its vector of distances to the set of chosen vertices. Let $$G_1$$ and $$G_2$$ be disjoint copies of a graph $$G$$, and let $$\sigma : V(G_1) \rightarrow V(G_2)$$ be a permutation. Then, a permutation graph$$G_{\sigma }=(V, E)$$ has the vertex set $$V=V(G_1) \cup V(G_2)$$ and the edge set $$E=E(G_1) \cup E(G_2) \cup \{uv \mid v=\sigma (u)\}$$. We show that $$2 \le \dim (G_{\sigma }) \le n-1$$ for any connected graph $$G$$ of order $$n$$ at least $$3$$. We give examples showing that neither is there a function $$f$$ such that $$\dim (G)<f(\dim (G_{\sigma }))$$ for all pairs $$(G,\sigma )$$, nor is there a function $$g$$ such that $$g(\dim (G))>\dim (G_{\sigma })$$ for all pairs $$(G, \sigma )$$. Further, we characterize permutation graphs $$G_{\sigma }$$ satisfying $$\dim (G_{\sigma })=n-1$$ when $$G$$ is a complete $$k$$-partite graph, a cycle, or a path on $$n$$ vertices. Metric dimension Permutation graph Complete -partite graph Cycle Path Kang, Cong X. aut Yi, Eunjeong aut Enthalten in Journal of combinatorial optimization Springer US, 1997 28(2013), 4 vom: 03. Jan., Seite 814-826 (DE-627)216539323 (DE-600)1339574-9 (DE-576)094421935 1382-6905 nnns volume:28 year:2013 number:4 day:03 month:01 pages:814-826 https://doi.org/10.1007/s10878-012-9587-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_24 GBV_ILN_26 GBV_ILN_70 GBV_ILN_2108 AR 28 2013 4 03 01 814-826 |
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| title_full |
On metric dimension of permutation graphs |
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Hallaway, Michael |
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Journal of combinatorial optimization |
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Journal of combinatorial optimization |
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eng |
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2013 |
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814 |
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Hallaway, Michael Kang, Cong X. Yi, Eunjeong |
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Hallaway, Michael |
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10.1007/s10878-012-9587-3 |
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510 |
| title_sort |
on metric dimension of permutation graphs |
| title_auth |
On metric dimension of permutation graphs |
| abstract |
Abstract The metric dimension$$\dim (G)$$ of a graph $$G$$ is the minimum number of vertices such that every vertex of $$G$$ is uniquely determined by its vector of distances to the set of chosen vertices. Let $$G_1$$ and $$G_2$$ be disjoint copies of a graph $$G$$, and let $$\sigma : V(G_1) \rightarrow V(G_2)$$ be a permutation. Then, a permutation graph$$G_{\sigma }=(V, E)$$ has the vertex set $$V=V(G_1) \cup V(G_2)$$ and the edge set $$E=E(G_1) \cup E(G_2) \cup \{uv \mid v=\sigma (u)\}$$. We show that $$2 \le \dim (G_{\sigma }) \le n-1$$ for any connected graph $$G$$ of order $$n$$ at least $$3$$. We give examples showing that neither is there a function $$f$$ such that $$\dim (G)<f(\dim (G_{\sigma }))$$ for all pairs $$(G,\sigma )$$, nor is there a function $$g$$ such that $$g(\dim (G))>\dim (G_{\sigma })$$ for all pairs $$(G, \sigma )$$. Further, we characterize permutation graphs $$G_{\sigma }$$ satisfying $$\dim (G_{\sigma })=n-1$$ when $$G$$ is a complete $$k$$-partite graph, a cycle, or a path on $$n$$ vertices. © Springer Science+Business Media New York 2012 |
| abstractGer |
Abstract The metric dimension$$\dim (G)$$ of a graph $$G$$ is the minimum number of vertices such that every vertex of $$G$$ is uniquely determined by its vector of distances to the set of chosen vertices. Let $$G_1$$ and $$G_2$$ be disjoint copies of a graph $$G$$, and let $$\sigma : V(G_1) \rightarrow V(G_2)$$ be a permutation. Then, a permutation graph$$G_{\sigma }=(V, E)$$ has the vertex set $$V=V(G_1) \cup V(G_2)$$ and the edge set $$E=E(G_1) \cup E(G_2) \cup \{uv \mid v=\sigma (u)\}$$. We show that $$2 \le \dim (G_{\sigma }) \le n-1$$ for any connected graph $$G$$ of order $$n$$ at least $$3$$. We give examples showing that neither is there a function $$f$$ such that $$\dim (G)<f(\dim (G_{\sigma }))$$ for all pairs $$(G,\sigma )$$, nor is there a function $$g$$ such that $$g(\dim (G))>\dim (G_{\sigma })$$ for all pairs $$(G, \sigma )$$. Further, we characterize permutation graphs $$G_{\sigma }$$ satisfying $$\dim (G_{\sigma })=n-1$$ when $$G$$ is a complete $$k$$-partite graph, a cycle, or a path on $$n$$ vertices. © Springer Science+Business Media New York 2012 |
| abstract_unstemmed |
Abstract The metric dimension$$\dim (G)$$ of a graph $$G$$ is the minimum number of vertices such that every vertex of $$G$$ is uniquely determined by its vector of distances to the set of chosen vertices. Let $$G_1$$ and $$G_2$$ be disjoint copies of a graph $$G$$, and let $$\sigma : V(G_1) \rightarrow V(G_2)$$ be a permutation. Then, a permutation graph$$G_{\sigma }=(V, E)$$ has the vertex set $$V=V(G_1) \cup V(G_2)$$ and the edge set $$E=E(G_1) \cup E(G_2) \cup \{uv \mid v=\sigma (u)\}$$. We show that $$2 \le \dim (G_{\sigma }) \le n-1$$ for any connected graph $$G$$ of order $$n$$ at least $$3$$. We give examples showing that neither is there a function $$f$$ such that $$\dim (G)<f(\dim (G_{\sigma }))$$ for all pairs $$(G,\sigma )$$, nor is there a function $$g$$ such that $$g(\dim (G))>\dim (G_{\sigma })$$ for all pairs $$(G, \sigma )$$. Further, we characterize permutation graphs $$G_{\sigma }$$ satisfying $$\dim (G_{\sigma })=n-1$$ when $$G$$ is a complete $$k$$-partite graph, a cycle, or a path on $$n$$ vertices. © Springer Science+Business Media New York 2012 |
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On metric dimension of permutation graphs |
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Kang, Cong X. Yi, Eunjeong |
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