The Approximate Rectangle of Influence Drawability Problem
Abstract Let ε1,ε2 be two non negative numbers. An approximate rectangle of influence drawing (also called (ε1,ε2)-RID) is a proximity drawing of a graph such that: (i) if u,v are adjacent vertices then their rectangle of influence “scaled down” by the factor $\frac{1}{1+\varepsilon_{1}}$ does not c...
Ausführliche Beschreibung
Autor*in: |
Di Giacomo, Emilio [verfasserIn] Liotta, Giuseppe [verfasserIn] Meijer, Henk [verfasserIn] |
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Format: |
E-Artikel |
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Sprache: |
Englisch |
Erschienen: |
2014 |
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Schlagwörter: |
Rectangle of influence drawings |
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Übergeordnetes Werk: |
Enthalten in: Algorithmica - New York, NY : Springer, 1986, 72(2014), 2 vom: 15. Jan., Seite 620-655 |
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Übergeordnetes Werk: |
volume:72 ; year:2014 ; number:2 ; day:15 ; month:01 ; pages:620-655 |
Links: |
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DOI / URN: |
10.1007/s00453-013-9866-0 |
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Katalog-ID: |
SPR006176348 |
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245 | 1 | 4 | |a The Approximate Rectangle of Influence Drawability Problem |
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520 | |a Abstract Let ε1,ε2 be two non negative numbers. An approximate rectangle of influence drawing (also called (ε1,ε2)-RID) is a proximity drawing of a graph such that: (i) if u,v are adjacent vertices then their rectangle of influence “scaled down” by the factor $\frac{1}{1+\varepsilon_{1}}$ does not contain other vertices of the drawing; (ii) if u,v are not adjacent, then their rectangle of influence “blown-up” by the factor 1+ε2 contains some vertices of the drawing other than u and v. Firstly, we prove that all planar graphs have an (ε1,ε2)-RID for any ε1>0 and ε2>0, while there exist planar graphs which do not admit an (ε1,0)-RID and planar graphs which do not admit a (0,ε2)-RID. Then, we investigate (0,ε2)-RIDs; we prove that both the outerplanar graphs and a suitably defined family of graphs without separating 3-cycles admit this type of drawing. Finally, we study polynomial area approximate rectangle of influence drawings and prove that (0,ε2)-RIDs of proper track planar graphs (a superclass of the outerplanar graphs) can be computed in polynomial area for any ε2>2. As for values of ε2 such that ε2≤2, we describe a drawing algorithm that computes (0,ε2)-RIDs of binary trees in area $O(n^{c + f(\varepsilon_{2})})$, where c is a positive constant, f(ε2) is a poly-logarithmic function that tends to infinity as ε2 tends to zero, and n is the number of vertices of the input tree. | ||
650 | 4 | |a Graph drawing |7 (dpeaa)DE-He213 | |
650 | 4 | |a Proximity drawings |7 (dpeaa)DE-He213 | |
650 | 4 | |a Rectangle of influence drawings |7 (dpeaa)DE-He213 | |
650 | 4 | |a Approximate proximity drawings |7 (dpeaa)DE-He213 | |
650 | 4 | |a Polynomial area approximation scheme |7 (dpeaa)DE-He213 | |
700 | 1 | |a Liotta, Giuseppe |e verfasserin |4 aut | |
700 | 1 | |a Meijer, Henk |e verfasserin |4 aut | |
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10.1007/s00453-013-9866-0 doi (DE-627)SPR006176348 (SPR)s00453-013-9866-0-e DE-627 ger DE-627 rakwb eng 004 ASE 510 004 ASE 54.00 bkl Di Giacomo, Emilio verfasserin aut The Approximate Rectangle of Influence Drawability Problem 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let ε1,ε2 be two non negative numbers. An approximate rectangle of influence drawing (also called (ε1,ε2)-RID) is a proximity drawing of a graph such that: (i) if u,v are adjacent vertices then their rectangle of influence “scaled down” by the factor $\frac{1}{1+\varepsilon_{1}}$ does not contain other vertices of the drawing; (ii) if u,v are not adjacent, then their rectangle of influence “blown-up” by the factor 1+ε2 contains some vertices of the drawing other than u and v. Firstly, we prove that all planar graphs have an (ε1,ε2)-RID for any ε1>0 and ε2>0, while there exist planar graphs which do not admit an (ε1,0)-RID and planar graphs which do not admit a (0,ε2)-RID. Then, we investigate (0,ε2)-RIDs; we prove that both the outerplanar graphs and a suitably defined family of graphs without separating 3-cycles admit this type of drawing. Finally, we study polynomial area approximate rectangle of influence drawings and prove that (0,ε2)-RIDs of proper track planar graphs (a superclass of the outerplanar graphs) can be computed in polynomial area for any ε2>2. As for values of ε2 such that ε2≤2, we describe a drawing algorithm that computes (0,ε2)-RIDs of binary trees in area $O(n^{c + f(\varepsilon_{2})})$, where c is a positive constant, f(ε2) is a poly-logarithmic function that tends to infinity as ε2 tends to zero, and n is the number of vertices of the input tree. Graph drawing (dpeaa)DE-He213 Proximity drawings (dpeaa)DE-He213 Rectangle of influence drawings (dpeaa)DE-He213 Approximate proximity drawings (dpeaa)DE-He213 Polynomial area approximation scheme (dpeaa)DE-He213 Liotta, Giuseppe verfasserin aut Meijer, Henk verfasserin aut Enthalten in Algorithmica New York, NY : Springer, 1986 72(2014), 2 vom: 15. Jan., Seite 620-655 (DE-627)253389704 (DE-600)1458414-1 1432-0541 nnns volume:72 year:2014 number:2 day:15 month:01 pages:620-655 https://dx.doi.org/10.1007/s00453-013-9866-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 54.00 ASE AR 72 2014 2 15 01 620-655 |
spelling |
10.1007/s00453-013-9866-0 doi (DE-627)SPR006176348 (SPR)s00453-013-9866-0-e DE-627 ger DE-627 rakwb eng 004 ASE 510 004 ASE 54.00 bkl Di Giacomo, Emilio verfasserin aut The Approximate Rectangle of Influence Drawability Problem 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let ε1,ε2 be two non negative numbers. An approximate rectangle of influence drawing (also called (ε1,ε2)-RID) is a proximity drawing of a graph such that: (i) if u,v are adjacent vertices then their rectangle of influence “scaled down” by the factor $\frac{1}{1+\varepsilon_{1}}$ does not contain other vertices of the drawing; (ii) if u,v are not adjacent, then their rectangle of influence “blown-up” by the factor 1+ε2 contains some vertices of the drawing other than u and v. Firstly, we prove that all planar graphs have an (ε1,ε2)-RID for any ε1>0 and ε2>0, while there exist planar graphs which do not admit an (ε1,0)-RID and planar graphs which do not admit a (0,ε2)-RID. Then, we investigate (0,ε2)-RIDs; we prove that both the outerplanar graphs and a suitably defined family of graphs without separating 3-cycles admit this type of drawing. Finally, we study polynomial area approximate rectangle of influence drawings and prove that (0,ε2)-RIDs of proper track planar graphs (a superclass of the outerplanar graphs) can be computed in polynomial area for any ε2>2. As for values of ε2 such that ε2≤2, we describe a drawing algorithm that computes (0,ε2)-RIDs of binary trees in area $O(n^{c + f(\varepsilon_{2})})$, where c is a positive constant, f(ε2) is a poly-logarithmic function that tends to infinity as ε2 tends to zero, and n is the number of vertices of the input tree. Graph drawing (dpeaa)DE-He213 Proximity drawings (dpeaa)DE-He213 Rectangle of influence drawings (dpeaa)DE-He213 Approximate proximity drawings (dpeaa)DE-He213 Polynomial area approximation scheme (dpeaa)DE-He213 Liotta, Giuseppe verfasserin aut Meijer, Henk verfasserin aut Enthalten in Algorithmica New York, NY : Springer, 1986 72(2014), 2 vom: 15. Jan., Seite 620-655 (DE-627)253389704 (DE-600)1458414-1 1432-0541 nnns volume:72 year:2014 number:2 day:15 month:01 pages:620-655 https://dx.doi.org/10.1007/s00453-013-9866-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 54.00 ASE AR 72 2014 2 15 01 620-655 |
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10.1007/s00453-013-9866-0 doi (DE-627)SPR006176348 (SPR)s00453-013-9866-0-e DE-627 ger DE-627 rakwb eng 004 ASE 510 004 ASE 54.00 bkl Di Giacomo, Emilio verfasserin aut The Approximate Rectangle of Influence Drawability Problem 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let ε1,ε2 be two non negative numbers. An approximate rectangle of influence drawing (also called (ε1,ε2)-RID) is a proximity drawing of a graph such that: (i) if u,v are adjacent vertices then their rectangle of influence “scaled down” by the factor $\frac{1}{1+\varepsilon_{1}}$ does not contain other vertices of the drawing; (ii) if u,v are not adjacent, then their rectangle of influence “blown-up” by the factor 1+ε2 contains some vertices of the drawing other than u and v. Firstly, we prove that all planar graphs have an (ε1,ε2)-RID for any ε1>0 and ε2>0, while there exist planar graphs which do not admit an (ε1,0)-RID and planar graphs which do not admit a (0,ε2)-RID. Then, we investigate (0,ε2)-RIDs; we prove that both the outerplanar graphs and a suitably defined family of graphs without separating 3-cycles admit this type of drawing. Finally, we study polynomial area approximate rectangle of influence drawings and prove that (0,ε2)-RIDs of proper track planar graphs (a superclass of the outerplanar graphs) can be computed in polynomial area for any ε2>2. As for values of ε2 such that ε2≤2, we describe a drawing algorithm that computes (0,ε2)-RIDs of binary trees in area $O(n^{c + f(\varepsilon_{2})})$, where c is a positive constant, f(ε2) is a poly-logarithmic function that tends to infinity as ε2 tends to zero, and n is the number of vertices of the input tree. Graph drawing (dpeaa)DE-He213 Proximity drawings (dpeaa)DE-He213 Rectangle of influence drawings (dpeaa)DE-He213 Approximate proximity drawings (dpeaa)DE-He213 Polynomial area approximation scheme (dpeaa)DE-He213 Liotta, Giuseppe verfasserin aut Meijer, Henk verfasserin aut Enthalten in Algorithmica New York, NY : Springer, 1986 72(2014), 2 vom: 15. Jan., Seite 620-655 (DE-627)253389704 (DE-600)1458414-1 1432-0541 nnns volume:72 year:2014 number:2 day:15 month:01 pages:620-655 https://dx.doi.org/10.1007/s00453-013-9866-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 54.00 ASE AR 72 2014 2 15 01 620-655 |
allfieldsGer |
10.1007/s00453-013-9866-0 doi (DE-627)SPR006176348 (SPR)s00453-013-9866-0-e DE-627 ger DE-627 rakwb eng 004 ASE 510 004 ASE 54.00 bkl Di Giacomo, Emilio verfasserin aut The Approximate Rectangle of Influence Drawability Problem 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let ε1,ε2 be two non negative numbers. An approximate rectangle of influence drawing (also called (ε1,ε2)-RID) is a proximity drawing of a graph such that: (i) if u,v are adjacent vertices then their rectangle of influence “scaled down” by the factor $\frac{1}{1+\varepsilon_{1}}$ does not contain other vertices of the drawing; (ii) if u,v are not adjacent, then their rectangle of influence “blown-up” by the factor 1+ε2 contains some vertices of the drawing other than u and v. Firstly, we prove that all planar graphs have an (ε1,ε2)-RID for any ε1>0 and ε2>0, while there exist planar graphs which do not admit an (ε1,0)-RID and planar graphs which do not admit a (0,ε2)-RID. Then, we investigate (0,ε2)-RIDs; we prove that both the outerplanar graphs and a suitably defined family of graphs without separating 3-cycles admit this type of drawing. Finally, we study polynomial area approximate rectangle of influence drawings and prove that (0,ε2)-RIDs of proper track planar graphs (a superclass of the outerplanar graphs) can be computed in polynomial area for any ε2>2. As for values of ε2 such that ε2≤2, we describe a drawing algorithm that computes (0,ε2)-RIDs of binary trees in area $O(n^{c + f(\varepsilon_{2})})$, where c is a positive constant, f(ε2) is a poly-logarithmic function that tends to infinity as ε2 tends to zero, and n is the number of vertices of the input tree. Graph drawing (dpeaa)DE-He213 Proximity drawings (dpeaa)DE-He213 Rectangle of influence drawings (dpeaa)DE-He213 Approximate proximity drawings (dpeaa)DE-He213 Polynomial area approximation scheme (dpeaa)DE-He213 Liotta, Giuseppe verfasserin aut Meijer, Henk verfasserin aut Enthalten in Algorithmica New York, NY : Springer, 1986 72(2014), 2 vom: 15. Jan., Seite 620-655 (DE-627)253389704 (DE-600)1458414-1 1432-0541 nnns volume:72 year:2014 number:2 day:15 month:01 pages:620-655 https://dx.doi.org/10.1007/s00453-013-9866-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 54.00 ASE AR 72 2014 2 15 01 620-655 |
allfieldsSound |
10.1007/s00453-013-9866-0 doi (DE-627)SPR006176348 (SPR)s00453-013-9866-0-e DE-627 ger DE-627 rakwb eng 004 ASE 510 004 ASE 54.00 bkl Di Giacomo, Emilio verfasserin aut The Approximate Rectangle of Influence Drawability Problem 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let ε1,ε2 be two non negative numbers. An approximate rectangle of influence drawing (also called (ε1,ε2)-RID) is a proximity drawing of a graph such that: (i) if u,v are adjacent vertices then their rectangle of influence “scaled down” by the factor $\frac{1}{1+\varepsilon_{1}}$ does not contain other vertices of the drawing; (ii) if u,v are not adjacent, then their rectangle of influence “blown-up” by the factor 1+ε2 contains some vertices of the drawing other than u and v. Firstly, we prove that all planar graphs have an (ε1,ε2)-RID for any ε1>0 and ε2>0, while there exist planar graphs which do not admit an (ε1,0)-RID and planar graphs which do not admit a (0,ε2)-RID. Then, we investigate (0,ε2)-RIDs; we prove that both the outerplanar graphs and a suitably defined family of graphs without separating 3-cycles admit this type of drawing. Finally, we study polynomial area approximate rectangle of influence drawings and prove that (0,ε2)-RIDs of proper track planar graphs (a superclass of the outerplanar graphs) can be computed in polynomial area for any ε2>2. As for values of ε2 such that ε2≤2, we describe a drawing algorithm that computes (0,ε2)-RIDs of binary trees in area $O(n^{c + f(\varepsilon_{2})})$, where c is a positive constant, f(ε2) is a poly-logarithmic function that tends to infinity as ε2 tends to zero, and n is the number of vertices of the input tree. Graph drawing (dpeaa)DE-He213 Proximity drawings (dpeaa)DE-He213 Rectangle of influence drawings (dpeaa)DE-He213 Approximate proximity drawings (dpeaa)DE-He213 Polynomial area approximation scheme (dpeaa)DE-He213 Liotta, Giuseppe verfasserin aut Meijer, Henk verfasserin aut Enthalten in Algorithmica New York, NY : Springer, 1986 72(2014), 2 vom: 15. Jan., Seite 620-655 (DE-627)253389704 (DE-600)1458414-1 1432-0541 nnns volume:72 year:2014 number:2 day:15 month:01 pages:620-655 https://dx.doi.org/10.1007/s00453-013-9866-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 54.00 ASE AR 72 2014 2 15 01 620-655 |
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Enthalten in Algorithmica 72(2014), 2 vom: 15. Jan., Seite 620-655 volume:72 year:2014 number:2 day:15 month:01 pages:620-655 |
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Graph drawing Proximity drawings Rectangle of influence drawings Approximate proximity drawings Polynomial area approximation scheme |
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Di Giacomo, Emilio @@aut@@ Liotta, Giuseppe @@aut@@ Meijer, Henk @@aut@@ |
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An approximate rectangle of influence drawing (also called (ε1,ε2)-RID) is a proximity drawing of a graph such that: (i) if u,v are adjacent vertices then their rectangle of influence “scaled down” by the factor $\frac{1}{1+\varepsilon_{1}}$ does not contain other vertices of the drawing; (ii) if u,v are not adjacent, then their rectangle of influence “blown-up” by the factor 1+ε2 contains some vertices of the drawing other than u and v. Firstly, we prove that all planar graphs have an (ε1,ε2)-RID for any ε1>0 and ε2>0, while there exist planar graphs which do not admit an (ε1,0)-RID and planar graphs which do not admit a (0,ε2)-RID. Then, we investigate (0,ε2)-RIDs; we prove that both the outerplanar graphs and a suitably defined family of graphs without separating 3-cycles admit this type of drawing. Finally, we study polynomial area approximate rectangle of influence drawings and prove that (0,ε2)-RIDs of proper track planar graphs (a superclass of the outerplanar graphs) can be computed in polynomial area for any ε2>2. As for values of ε2 such that ε2≤2, we describe a drawing algorithm that computes (0,ε2)-RIDs of binary trees in area $O(n^{c + f(\varepsilon_{2})})$, where c is a positive constant, f(ε2) is a poly-logarithmic function that tends to infinity as ε2 tends to zero, and n is the number of vertices of the input tree.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Graph drawing</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Proximity drawings</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Rectangle of influence drawings</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Approximate proximity drawings</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Polynomial area approximation scheme</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Liotta, Giuseppe</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Meijer, Henk</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Algorithmica</subfield><subfield code="d">New York, NY : Springer, 1986</subfield><subfield code="g">72(2014), 2 vom: 15. 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Di Giacomo, Emilio |
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Di Giacomo, Emilio ddc 004 ddc 510 bkl 54.00 misc Graph drawing misc Proximity drawings misc Rectangle of influence drawings misc Approximate proximity drawings misc Polynomial area approximation scheme The Approximate Rectangle of Influence Drawability Problem |
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004 ASE 510 004 ASE 54.00 bkl The Approximate Rectangle of Influence Drawability Problem Graph drawing (dpeaa)DE-He213 Proximity drawings (dpeaa)DE-He213 Rectangle of influence drawings (dpeaa)DE-He213 Approximate proximity drawings (dpeaa)DE-He213 Polynomial area approximation scheme (dpeaa)DE-He213 |
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ddc 004 ddc 510 bkl 54.00 misc Graph drawing misc Proximity drawings misc Rectangle of influence drawings misc Approximate proximity drawings misc Polynomial area approximation scheme |
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The Approximate Rectangle of Influence Drawability Problem |
abstract |
Abstract Let ε1,ε2 be two non negative numbers. An approximate rectangle of influence drawing (also called (ε1,ε2)-RID) is a proximity drawing of a graph such that: (i) if u,v are adjacent vertices then their rectangle of influence “scaled down” by the factor $\frac{1}{1+\varepsilon_{1}}$ does not contain other vertices of the drawing; (ii) if u,v are not adjacent, then their rectangle of influence “blown-up” by the factor 1+ε2 contains some vertices of the drawing other than u and v. Firstly, we prove that all planar graphs have an (ε1,ε2)-RID for any ε1>0 and ε2>0, while there exist planar graphs which do not admit an (ε1,0)-RID and planar graphs which do not admit a (0,ε2)-RID. Then, we investigate (0,ε2)-RIDs; we prove that both the outerplanar graphs and a suitably defined family of graphs without separating 3-cycles admit this type of drawing. Finally, we study polynomial area approximate rectangle of influence drawings and prove that (0,ε2)-RIDs of proper track planar graphs (a superclass of the outerplanar graphs) can be computed in polynomial area for any ε2>2. As for values of ε2 such that ε2≤2, we describe a drawing algorithm that computes (0,ε2)-RIDs of binary trees in area $O(n^{c + f(\varepsilon_{2})})$, where c is a positive constant, f(ε2) is a poly-logarithmic function that tends to infinity as ε2 tends to zero, and n is the number of vertices of the input tree. |
abstractGer |
Abstract Let ε1,ε2 be two non negative numbers. An approximate rectangle of influence drawing (also called (ε1,ε2)-RID) is a proximity drawing of a graph such that: (i) if u,v are adjacent vertices then their rectangle of influence “scaled down” by the factor $\frac{1}{1+\varepsilon_{1}}$ does not contain other vertices of the drawing; (ii) if u,v are not adjacent, then their rectangle of influence “blown-up” by the factor 1+ε2 contains some vertices of the drawing other than u and v. Firstly, we prove that all planar graphs have an (ε1,ε2)-RID for any ε1>0 and ε2>0, while there exist planar graphs which do not admit an (ε1,0)-RID and planar graphs which do not admit a (0,ε2)-RID. Then, we investigate (0,ε2)-RIDs; we prove that both the outerplanar graphs and a suitably defined family of graphs without separating 3-cycles admit this type of drawing. Finally, we study polynomial area approximate rectangle of influence drawings and prove that (0,ε2)-RIDs of proper track planar graphs (a superclass of the outerplanar graphs) can be computed in polynomial area for any ε2>2. As for values of ε2 such that ε2≤2, we describe a drawing algorithm that computes (0,ε2)-RIDs of binary trees in area $O(n^{c + f(\varepsilon_{2})})$, where c is a positive constant, f(ε2) is a poly-logarithmic function that tends to infinity as ε2 tends to zero, and n is the number of vertices of the input tree. |
abstract_unstemmed |
Abstract Let ε1,ε2 be two non negative numbers. An approximate rectangle of influence drawing (also called (ε1,ε2)-RID) is a proximity drawing of a graph such that: (i) if u,v are adjacent vertices then their rectangle of influence “scaled down” by the factor $\frac{1}{1+\varepsilon_{1}}$ does not contain other vertices of the drawing; (ii) if u,v are not adjacent, then their rectangle of influence “blown-up” by the factor 1+ε2 contains some vertices of the drawing other than u and v. Firstly, we prove that all planar graphs have an (ε1,ε2)-RID for any ε1>0 and ε2>0, while there exist planar graphs which do not admit an (ε1,0)-RID and planar graphs which do not admit a (0,ε2)-RID. Then, we investigate (0,ε2)-RIDs; we prove that both the outerplanar graphs and a suitably defined family of graphs without separating 3-cycles admit this type of drawing. Finally, we study polynomial area approximate rectangle of influence drawings and prove that (0,ε2)-RIDs of proper track planar graphs (a superclass of the outerplanar graphs) can be computed in polynomial area for any ε2>2. As for values of ε2 such that ε2≤2, we describe a drawing algorithm that computes (0,ε2)-RIDs of binary trees in area $O(n^{c + f(\varepsilon_{2})})$, where c is a positive constant, f(ε2) is a poly-logarithmic function that tends to infinity as ε2 tends to zero, and n is the number of vertices of the input tree. |
collection_details |
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container_issue |
2 |
title_short |
The Approximate Rectangle of Influence Drawability Problem |
url |
https://dx.doi.org/10.1007/s00453-013-9866-0 |
remote_bool |
true |
author2 |
Liotta, Giuseppe Meijer, Henk |
author2Str |
Liotta, Giuseppe Meijer, Henk |
ppnlink |
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isOA_txt |
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hochschulschrift_bool |
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doi_str |
10.1007/s00453-013-9866-0 |
up_date |
2024-07-03T21:21:09.913Z |
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score |
7.3993006 |