The connected metric dimension at a vertex of a graph
The notion of metric dimension, dim ( G ) , of a graph G, as well as a number of variants, is now well studied. In this paper, we begin a local analysis of this notion by introducing...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Eroh, Linda [verfasserIn] Kang, Cong X. [verfasserIn] Yi, Eunjeong [verfasserIn] |
|---|
| Format: |
E-Artikel |
|---|---|
| Sprache: |
Englisch |
| Erschienen: |
2018 |
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| Schlagwörter: |
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| Übergeordnetes Werk: |
Enthalten in: Theoretical computer science - Amsterdam [u.a.] : Elsevier, 1975, 806, Seite 53-69 |
|---|---|
| Übergeordnetes Werk: |
volume:806 ; pages:53-69 |
| DOI / URN: |
10.1016/j.tcs.2018.11.002 |
|---|
| Katalog-ID: |
ELV003454630 |
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| 245 | 1 | 0 | |a The connected metric dimension at a vertex of a graph |
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| 520 | |a The notion of metric dimension, dim ( G ) , of a graph G, as well as a number of variants, is now well studied. In this paper, we begin a local analysis of this notion by introducing cdim G ( v ) , the connected metric dimension of G at a vertex v, which is defined as follows: a set of vertices S of G is a resolving set if, for any pair of distinct vertices x and y of G, there is a vertex z ∈ S such that the distance between z and x is distinct from the distance between z and y in G. We say that a resolving set S is connected if S induces a connected subgraph of G. Then, cdim G ( v ) is defined to be the minimum of the cardinalities of all connected resolving sets which contain the vertex v. The connected metric dimension of G, denoted by cdim ( G ) , is min { cdim G ( v ) : v ∈ V ( G ) } . Noting that 1 ≤ dim ( G ) ≤ cdim ( G ) ≤ cdim G ( v ) ≤ | V ( G ) | − 1 for any vertex v of G, we show the existence of a pair ( G , v ) such that cdim G ( v ) takes all positive integer values from dim ( G ) to | V ( G ) | − 1 , as v varies in a fixed graph G. We characterize graphs G and their vertices v satisfying cdim G ( v ) ∈ { 1 , | V ( G ) | − 1 } . We show that cdim ( G ) = 2 implies G is planar, whereas it is well known that there is a non-planar graph H with dim ( H ) = 2 . We also characterize trees and unicyclic graphs G satisfying cdim ( G ) = dim ( G ) . We show that cdim ( G ) − dim ( G ) can be arbitrarily large. We determine cdim ( G ) and cdim G ( v ) for some classes of graphs. We further examine the effect of vertex or edge deletion on the connected metric dimension. We conclude with some open problems. | ||
| 650 | 4 | |a Distance | |
| 650 | 4 | |a Connected resolving set | |
| 650 | 4 | |a Connected metric dimension | |
| 650 | 4 | |a Metric dimension | |
| 700 | 1 | |a Kang, Cong X. |e verfasserin |4 aut | |
| 700 | 1 | |a Yi, Eunjeong |e verfasserin |4 aut | |
| 773 | 0 | 8 | |i Enthalten in |t Theoretical computer science |d Amsterdam [u.a.] : Elsevier, 1975 |g 806, Seite 53-69 |h Online-Ressource |w (DE-627)265784174 |w (DE-600)1466347-8 |w (DE-576)074891030 |7 nnns |
| 773 | 1 | 8 | |g volume:806 |g pages:53-69 |
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| 936 | b | k | |a 54.10 |j Theoretische Informatik |
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10.1016/j.tcs.2018.11.002 doi (DE-627)ELV003454630 (ELSEVIER)S0304-3975(18)30667-4 DE-627 ger DE-627 rda eng 004 DE-600 54.10 bkl Eroh, Linda verfasserin aut The connected metric dimension at a vertex of a graph 2018 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The notion of metric dimension, dim ( G ) , of a graph G, as well as a number of variants, is now well studied. In this paper, we begin a local analysis of this notion by introducing cdim G ( v ) , the connected metric dimension of G at a vertex v, which is defined as follows: a set of vertices S of G is a resolving set if, for any pair of distinct vertices x and y of G, there is a vertex z ∈ S such that the distance between z and x is distinct from the distance between z and y in G. We say that a resolving set S is connected if S induces a connected subgraph of G. Then, cdim G ( v ) is defined to be the minimum of the cardinalities of all connected resolving sets which contain the vertex v. The connected metric dimension of G, denoted by cdim ( G ) , is min { cdim G ( v ) : v ∈ V ( G ) } . Noting that 1 ≤ dim ( G ) ≤ cdim ( G ) ≤ cdim G ( v ) ≤ | V ( G ) | − 1 for any vertex v of G, we show the existence of a pair ( G , v ) such that cdim G ( v ) takes all positive integer values from dim ( G ) to | V ( G ) | − 1 , as v varies in a fixed graph G. We characterize graphs G and their vertices v satisfying cdim G ( v ) ∈ { 1 , | V ( G ) | − 1 } . We show that cdim ( G ) = 2 implies G is planar, whereas it is well known that there is a non-planar graph H with dim ( H ) = 2 . We also characterize trees and unicyclic graphs G satisfying cdim ( G ) = dim ( G ) . We show that cdim ( G ) − dim ( G ) can be arbitrarily large. We determine cdim ( G ) and cdim G ( v ) for some classes of graphs. We further examine the effect of vertex or edge deletion on the connected metric dimension. We conclude with some open problems. Distance Connected resolving set Connected metric dimension Metric dimension Kang, Cong X. verfasserin aut Yi, Eunjeong verfasserin aut Enthalten in Theoretical computer science Amsterdam [u.a.] : Elsevier, 1975 806, Seite 53-69 Online-Ressource (DE-627)265784174 (DE-600)1466347-8 (DE-576)074891030 nnns volume:806 pages:53-69 GBV_USEFLAG_U SYSFLAG_U GBV_ELV GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 54.10 Theoretische Informatik AR 806 53-69 |
| spelling |
10.1016/j.tcs.2018.11.002 doi (DE-627)ELV003454630 (ELSEVIER)S0304-3975(18)30667-4 DE-627 ger DE-627 rda eng 004 DE-600 54.10 bkl Eroh, Linda verfasserin aut The connected metric dimension at a vertex of a graph 2018 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The notion of metric dimension, dim ( G ) , of a graph G, as well as a number of variants, is now well studied. In this paper, we begin a local analysis of this notion by introducing cdim G ( v ) , the connected metric dimension of G at a vertex v, which is defined as follows: a set of vertices S of G is a resolving set if, for any pair of distinct vertices x and y of G, there is a vertex z ∈ S such that the distance between z and x is distinct from the distance between z and y in G. We say that a resolving set S is connected if S induces a connected subgraph of G. Then, cdim G ( v ) is defined to be the minimum of the cardinalities of all connected resolving sets which contain the vertex v. The connected metric dimension of G, denoted by cdim ( G ) , is min { cdim G ( v ) : v ∈ V ( G ) } . Noting that 1 ≤ dim ( G ) ≤ cdim ( G ) ≤ cdim G ( v ) ≤ | V ( G ) | − 1 for any vertex v of G, we show the existence of a pair ( G , v ) such that cdim G ( v ) takes all positive integer values from dim ( G ) to | V ( G ) | − 1 , as v varies in a fixed graph G. We characterize graphs G and their vertices v satisfying cdim G ( v ) ∈ { 1 , | V ( G ) | − 1 } . We show that cdim ( G ) = 2 implies G is planar, whereas it is well known that there is a non-planar graph H with dim ( H ) = 2 . We also characterize trees and unicyclic graphs G satisfying cdim ( G ) = dim ( G ) . We show that cdim ( G ) − dim ( G ) can be arbitrarily large. We determine cdim ( G ) and cdim G ( v ) for some classes of graphs. We further examine the effect of vertex or edge deletion on the connected metric dimension. We conclude with some open problems. Distance Connected resolving set Connected metric dimension Metric dimension Kang, Cong X. verfasserin aut Yi, Eunjeong verfasserin aut Enthalten in Theoretical computer science Amsterdam [u.a.] : Elsevier, 1975 806, Seite 53-69 Online-Ressource (DE-627)265784174 (DE-600)1466347-8 (DE-576)074891030 nnns volume:806 pages:53-69 GBV_USEFLAG_U SYSFLAG_U GBV_ELV GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 54.10 Theoretische Informatik AR 806 53-69 |
| allfields_unstemmed |
10.1016/j.tcs.2018.11.002 doi (DE-627)ELV003454630 (ELSEVIER)S0304-3975(18)30667-4 DE-627 ger DE-627 rda eng 004 DE-600 54.10 bkl Eroh, Linda verfasserin aut The connected metric dimension at a vertex of a graph 2018 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The notion of metric dimension, dim ( G ) , of a graph G, as well as a number of variants, is now well studied. In this paper, we begin a local analysis of this notion by introducing cdim G ( v ) , the connected metric dimension of G at a vertex v, which is defined as follows: a set of vertices S of G is a resolving set if, for any pair of distinct vertices x and y of G, there is a vertex z ∈ S such that the distance between z and x is distinct from the distance between z and y in G. We say that a resolving set S is connected if S induces a connected subgraph of G. Then, cdim G ( v ) is defined to be the minimum of the cardinalities of all connected resolving sets which contain the vertex v. The connected metric dimension of G, denoted by cdim ( G ) , is min { cdim G ( v ) : v ∈ V ( G ) } . Noting that 1 ≤ dim ( G ) ≤ cdim ( G ) ≤ cdim G ( v ) ≤ | V ( G ) | − 1 for any vertex v of G, we show the existence of a pair ( G , v ) such that cdim G ( v ) takes all positive integer values from dim ( G ) to | V ( G ) | − 1 , as v varies in a fixed graph G. We characterize graphs G and their vertices v satisfying cdim G ( v ) ∈ { 1 , | V ( G ) | − 1 } . We show that cdim ( G ) = 2 implies G is planar, whereas it is well known that there is a non-planar graph H with dim ( H ) = 2 . We also characterize trees and unicyclic graphs G satisfying cdim ( G ) = dim ( G ) . We show that cdim ( G ) − dim ( G ) can be arbitrarily large. We determine cdim ( G ) and cdim G ( v ) for some classes of graphs. We further examine the effect of vertex or edge deletion on the connected metric dimension. We conclude with some open problems. Distance Connected resolving set Connected metric dimension Metric dimension Kang, Cong X. verfasserin aut Yi, Eunjeong verfasserin aut Enthalten in Theoretical computer science Amsterdam [u.a.] : Elsevier, 1975 806, Seite 53-69 Online-Ressource (DE-627)265784174 (DE-600)1466347-8 (DE-576)074891030 nnns volume:806 pages:53-69 GBV_USEFLAG_U SYSFLAG_U GBV_ELV GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 54.10 Theoretische Informatik AR 806 53-69 |
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10.1016/j.tcs.2018.11.002 doi (DE-627)ELV003454630 (ELSEVIER)S0304-3975(18)30667-4 DE-627 ger DE-627 rda eng 004 DE-600 54.10 bkl Eroh, Linda verfasserin aut The connected metric dimension at a vertex of a graph 2018 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The notion of metric dimension, dim ( G ) , of a graph G, as well as a number of variants, is now well studied. In this paper, we begin a local analysis of this notion by introducing cdim G ( v ) , the connected metric dimension of G at a vertex v, which is defined as follows: a set of vertices S of G is a resolving set if, for any pair of distinct vertices x and y of G, there is a vertex z ∈ S such that the distance between z and x is distinct from the distance between z and y in G. We say that a resolving set S is connected if S induces a connected subgraph of G. Then, cdim G ( v ) is defined to be the minimum of the cardinalities of all connected resolving sets which contain the vertex v. The connected metric dimension of G, denoted by cdim ( G ) , is min { cdim G ( v ) : v ∈ V ( G ) } . Noting that 1 ≤ dim ( G ) ≤ cdim ( G ) ≤ cdim G ( v ) ≤ | V ( G ) | − 1 for any vertex v of G, we show the existence of a pair ( G , v ) such that cdim G ( v ) takes all positive integer values from dim ( G ) to | V ( G ) | − 1 , as v varies in a fixed graph G. We characterize graphs G and their vertices v satisfying cdim G ( v ) ∈ { 1 , | V ( G ) | − 1 } . We show that cdim ( G ) = 2 implies G is planar, whereas it is well known that there is a non-planar graph H with dim ( H ) = 2 . We also characterize trees and unicyclic graphs G satisfying cdim ( G ) = dim ( G ) . We show that cdim ( G ) − dim ( G ) can be arbitrarily large. We determine cdim ( G ) and cdim G ( v ) for some classes of graphs. We further examine the effect of vertex or edge deletion on the connected metric dimension. We conclude with some open problems. Distance Connected resolving set Connected metric dimension Metric dimension Kang, Cong X. verfasserin aut Yi, Eunjeong verfasserin aut Enthalten in Theoretical computer science Amsterdam [u.a.] : Elsevier, 1975 806, Seite 53-69 Online-Ressource (DE-627)265784174 (DE-600)1466347-8 (DE-576)074891030 nnns volume:806 pages:53-69 GBV_USEFLAG_U SYSFLAG_U GBV_ELV GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 54.10 Theoretische Informatik AR 806 53-69 |
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10.1016/j.tcs.2018.11.002 doi (DE-627)ELV003454630 (ELSEVIER)S0304-3975(18)30667-4 DE-627 ger DE-627 rda eng 004 DE-600 54.10 bkl Eroh, Linda verfasserin aut The connected metric dimension at a vertex of a graph 2018 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The notion of metric dimension, dim ( G ) , of a graph G, as well as a number of variants, is now well studied. In this paper, we begin a local analysis of this notion by introducing cdim G ( v ) , the connected metric dimension of G at a vertex v, which is defined as follows: a set of vertices S of G is a resolving set if, for any pair of distinct vertices x and y of G, there is a vertex z ∈ S such that the distance between z and x is distinct from the distance between z and y in G. We say that a resolving set S is connected if S induces a connected subgraph of G. Then, cdim G ( v ) is defined to be the minimum of the cardinalities of all connected resolving sets which contain the vertex v. The connected metric dimension of G, denoted by cdim ( G ) , is min { cdim G ( v ) : v ∈ V ( G ) } . Noting that 1 ≤ dim ( G ) ≤ cdim ( G ) ≤ cdim G ( v ) ≤ | V ( G ) | − 1 for any vertex v of G, we show the existence of a pair ( G , v ) such that cdim G ( v ) takes all positive integer values from dim ( G ) to | V ( G ) | − 1 , as v varies in a fixed graph G. We characterize graphs G and their vertices v satisfying cdim G ( v ) ∈ { 1 , | V ( G ) | − 1 } . We show that cdim ( G ) = 2 implies G is planar, whereas it is well known that there is a non-planar graph H with dim ( H ) = 2 . We also characterize trees and unicyclic graphs G satisfying cdim ( G ) = dim ( G ) . We show that cdim ( G ) − dim ( G ) can be arbitrarily large. We determine cdim ( G ) and cdim G ( v ) for some classes of graphs. We further examine the effect of vertex or edge deletion on the connected metric dimension. We conclude with some open problems. Distance Connected resolving set Connected metric dimension Metric dimension Kang, Cong X. verfasserin aut Yi, Eunjeong verfasserin aut Enthalten in Theoretical computer science Amsterdam [u.a.] : Elsevier, 1975 806, Seite 53-69 Online-Ressource (DE-627)265784174 (DE-600)1466347-8 (DE-576)074891030 nnns volume:806 pages:53-69 GBV_USEFLAG_U SYSFLAG_U GBV_ELV GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 54.10 Theoretische Informatik AR 806 53-69 |
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the connected metric dimension at a vertex of a graph |
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The connected metric dimension at a vertex of a graph |
| abstract |
The notion of metric dimension, dim ( G ) , of a graph G, as well as a number of variants, is now well studied. In this paper, we begin a local analysis of this notion by introducing cdim G ( v ) , the connected metric dimension of G at a vertex v, which is defined as follows: a set of vertices S of G is a resolving set if, for any pair of distinct vertices x and y of G, there is a vertex z ∈ S such that the distance between z and x is distinct from the distance between z and y in G. We say that a resolving set S is connected if S induces a connected subgraph of G. Then, cdim G ( v ) is defined to be the minimum of the cardinalities of all connected resolving sets which contain the vertex v. The connected metric dimension of G, denoted by cdim ( G ) , is min { cdim G ( v ) : v ∈ V ( G ) } . Noting that 1 ≤ dim ( G ) ≤ cdim ( G ) ≤ cdim G ( v ) ≤ | V ( G ) | − 1 for any vertex v of G, we show the existence of a pair ( G , v ) such that cdim G ( v ) takes all positive integer values from dim ( G ) to | V ( G ) | − 1 , as v varies in a fixed graph G. We characterize graphs G and their vertices v satisfying cdim G ( v ) ∈ { 1 , | V ( G ) | − 1 } . We show that cdim ( G ) = 2 implies G is planar, whereas it is well known that there is a non-planar graph H with dim ( H ) = 2 . We also characterize trees and unicyclic graphs G satisfying cdim ( G ) = dim ( G ) . We show that cdim ( G ) − dim ( G ) can be arbitrarily large. We determine cdim ( G ) and cdim G ( v ) for some classes of graphs. We further examine the effect of vertex or edge deletion on the connected metric dimension. We conclude with some open problems. |
| abstractGer |
The notion of metric dimension, dim ( G ) , of a graph G, as well as a number of variants, is now well studied. In this paper, we begin a local analysis of this notion by introducing cdim G ( v ) , the connected metric dimension of G at a vertex v, which is defined as follows: a set of vertices S of G is a resolving set if, for any pair of distinct vertices x and y of G, there is a vertex z ∈ S such that the distance between z and x is distinct from the distance between z and y in G. We say that a resolving set S is connected if S induces a connected subgraph of G. Then, cdim G ( v ) is defined to be the minimum of the cardinalities of all connected resolving sets which contain the vertex v. The connected metric dimension of G, denoted by cdim ( G ) , is min { cdim G ( v ) : v ∈ V ( G ) } . Noting that 1 ≤ dim ( G ) ≤ cdim ( G ) ≤ cdim G ( v ) ≤ | V ( G ) | − 1 for any vertex v of G, we show the existence of a pair ( G , v ) such that cdim G ( v ) takes all positive integer values from dim ( G ) to | V ( G ) | − 1 , as v varies in a fixed graph G. We characterize graphs G and their vertices v satisfying cdim G ( v ) ∈ { 1 , | V ( G ) | − 1 } . We show that cdim ( G ) = 2 implies G is planar, whereas it is well known that there is a non-planar graph H with dim ( H ) = 2 . We also characterize trees and unicyclic graphs G satisfying cdim ( G ) = dim ( G ) . We show that cdim ( G ) − dim ( G ) can be arbitrarily large. We determine cdim ( G ) and cdim G ( v ) for some classes of graphs. We further examine the effect of vertex or edge deletion on the connected metric dimension. We conclude with some open problems. |
| abstract_unstemmed |
The notion of metric dimension, dim ( G ) , of a graph G, as well as a number of variants, is now well studied. In this paper, we begin a local analysis of this notion by introducing cdim G ( v ) , the connected metric dimension of G at a vertex v, which is defined as follows: a set of vertices S of G is a resolving set if, for any pair of distinct vertices x and y of G, there is a vertex z ∈ S such that the distance between z and x is distinct from the distance between z and y in G. We say that a resolving set S is connected if S induces a connected subgraph of G. Then, cdim G ( v ) is defined to be the minimum of the cardinalities of all connected resolving sets which contain the vertex v. The connected metric dimension of G, denoted by cdim ( G ) , is min { cdim G ( v ) : v ∈ V ( G ) } . Noting that 1 ≤ dim ( G ) ≤ cdim ( G ) ≤ cdim G ( v ) ≤ | V ( G ) | − 1 for any vertex v of G, we show the existence of a pair ( G , v ) such that cdim G ( v ) takes all positive integer values from dim ( G ) to | V ( G ) | − 1 , as v varies in a fixed graph G. We characterize graphs G and their vertices v satisfying cdim G ( v ) ∈ { 1 , | V ( G ) | − 1 } . We show that cdim ( G ) = 2 implies G is planar, whereas it is well known that there is a non-planar graph H with dim ( H ) = 2 . We also characterize trees and unicyclic graphs G satisfying cdim ( G ) = dim ( G ) . We show that cdim ( G ) − dim ( G ) can be arbitrarily large. We determine cdim ( G ) and cdim G ( v ) for some classes of graphs. We further examine the effect of vertex or edge deletion on the connected metric dimension. We conclude with some open problems. |
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