Straight Line Triangle Representations
A straight line triangle representation (SLTR) of a planar graph is a straight line drawing such that all the faces including the outer face have triangular shape. Such a drawing can be viewed as a tiling of a triangle using triangles with the input graph as skeletal structure. In this paper we pres...
Saved in:
| Published in | Discrete & computational geometry Vol. 57; no. 2; pp. 257 - 280 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.03.2017
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0179-5376 1432-0444 |
| DOI | 10.1007/s00454-016-9850-y |
Cover
| Abstract | A straight line triangle representation (SLTR) of a planar graph is a straight line drawing such that all the faces including the outer face have triangular shape. Such a drawing can be viewed as a tiling of a triangle using triangles with the input graph as skeletal structure. In this paper we present a characterization of graphs that have an SLTR. The characterization is based on flat angle assignments, i.e., selections of angles of the graph that have size
π
in the representation. We also provide a second characterization in terms of contact systems of pseudosegments. With the aid of discrete harmonic functions we show that contact systems of pseudosegments that respect certain conditions are stretchable. The stretching procedure is then used to get straight line triangle representations. Since the discrete harmonic function approach is quite flexible it allows further applications, we mention some of them. The drawback of the characterization of SLTRs is that we are not able to effectively check whether a given graph admits a flat angle assignment that fulfills the conditions. Hence it is still open to decide whether the recognition of graphs that admit straight line triangle representation is polynomially tractable. |
|---|---|
| AbstractList | (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).A straight line triangle representation (SLTR) of a planar graph is a straight line drawing such that all the faces including the outer face have triangular shape. Such a drawing can be viewed as a tiling of a triangle using triangles with the input graph as skeletal structure. In this paper we present a characterization of graphs that have an SLTR. The characterization is based on flat angle assignments, i.e., selections of angles of the graph that have size ... in the representation. We also provide a second characterization in terms of contact systems of pseudosegments. With the aid of discrete harmonic functions we show that contact systems of pseudosegments that respect certain conditions are stretchable. The stretching procedure is then used to get straight line triangle representations. Since the discrete harmonic function approach is quite flexible it allows further applications, we mention some of them. The drawback of the characterization of SLTRs is that we are not able to effectively check whether a given graph admits a flat angle assignment that fulfills the conditions. Hence it is still open to decide whether the recognition of graphs that admit straight line triangle representation is polynomially tractable. A straight line triangle representation (SLTR) of a planar graph is a straight line drawing such that all the faces including the outer face have triangular shape. Such a drawing can be viewed as a tiling of a triangle using triangles with the input graph as skeletal structure. In this paper we present a characterization of graphs that have an SLTR. The characterization is based on flat angle assignments, i.e., selections of angles of the graph that have size π in the representation. We also provide a second characterization in terms of contact systems of pseudosegments. With the aid of discrete harmonic functions we show that contact systems of pseudosegments that respect certain conditions are stretchable. The stretching procedure is then used to get straight line triangle representations. Since the discrete harmonic function approach is quite flexible it allows further applications, we mention some of them. The drawback of the characterization of SLTRs is that we are not able to effectively check whether a given graph admits a flat angle assignment that fulfills the conditions. Hence it is still open to decide whether the recognition of graphs that admit straight line triangle representation is polynomially tractable. |
| Author | Felsner, Stefan Aerts, Nieke |
| Author_xml | – sequence: 1 givenname: Nieke orcidid: 0000-0003-2702-5964 surname: Aerts fullname: Aerts, Nieke email: nkmaerts@gmail.com organization: Institut für Mathematik, Technische Universität Berlin – sequence: 2 givenname: Stefan surname: Felsner fullname: Felsner, Stefan organization: Institut für Mathematik, Technische Universität Berlin |
| BookMark | eNp9kM1KAzEYRYNUsK0-gLuCIG6i-U9mKcU_KAha1yGTydSUaaYmM4u-vRnGhRR09W3OudzvzsAktMEBcInRLUZI3iWEGGcQYQELxRE8nIApZpRAxBibgCnCsoCcSnEGZiltUcYLpKbg-r2Lxm8-u8XKB7dYR2_CpnGLN7ePLrnQmc63IZ2D09o0yV383Dn4eHxYL5_h6vXpZXm_gpayooPKcGWsrHipjMIES0WrmkmKS4GExYQUleMFVlVRqpLWtWWG1IQqYoSoFCJ0Dm7G3H1sv3qXOr3zybqmMcG1fdJYKZZjhMAZvTpCt20fQ26XKcEJz3kDJUfKxjal6Gpt_fjT8HejMdLDfnrcT-f99LCfPmQTH5n76HcmHv51yOikzIaNi786_Sl9Ax5fgs0 |
| CODEN | DCGEER |
| CitedBy_id | crossref_primary_10_1109_TNSE_2018_2856280 |
| Cites_doi | 10.1007/978-3-319-03841-4_14 10.1016/j.comgeo.2004.07.003 10.1023/B:ORDE.0000009251.68514.8b 10.1023/A:1010604726900 10.1007/978-3-642-36763-2_18 10.4171/dm/117 10.1007/978-88-7642-475-5_80 10.1007/978-3-642-18469-7_23 10.1016/j.dam.2005.12.009 10.1017/S0963548300001139 10.1007/s00454-012-9400-1 10.1007/BF00353652 10.7155/jgaa.00372 10.1016/j.jctb.2004.07.001 10.1112/plms/s3-13.1.743 10.1007/s00454-007-9027-9 |
| ContentType | Journal Article |
| Copyright | Springer Science+Business Media New York 2017 Discrete & Computational Geometry is a copyright of Springer, 2017. |
| Copyright_xml | – notice: Springer Science+Business Media New York 2017 – notice: Discrete & Computational Geometry is a copyright of Springer, 2017. |
| DBID | AAYXX CITATION 3V. 7SC 7TB 7XB 88I 8AL 8AO 8FD 8FE 8FG 8FK 8G5 ABJCF ABUWG AFKRA ARAPS AZQEC BENPR BGLVJ CCPQU DWQXO FR3 GNUQQ GUQSH HCIFZ JQ2 K7- KR7 L6V L7M L~C L~D M0N M2O M2P M7S MBDVC P5Z P62 PADUT PHGZM PHGZT PKEHL PQEST PQGLB PQQKQ PQUKI PRINS PTHSS Q9U |
| DOI | 10.1007/s00454-016-9850-y |
| DatabaseName | CrossRef ProQuest Central (Corporate) Computer and Information Systems Abstracts Mechanical & Transportation Engineering Abstracts ProQuest Central (purchase pre-March 2016) Science Database (Alumni Edition) Computing Database (Alumni Edition) ProQuest Pharma Collection Technology Research Database ProQuest SciTech Collection ProQuest Technology Collection ProQuest Central (Alumni) (purchase pre-March 2016) ProQuest Research Library ProQuest Materials Science & Engineering ProQuest Central (Alumni) ProQuest Central UK/Ireland Advanced Technologies & Aerospace Collection ProQuest Central Essentials - QC ProQuest Central Technology Collection (via ProQuest SciTech Premium Collection) ProQuest One Community College ProQuest Central Engineering Research Database ProQuest Central Student ProQuest Research Library SciTech Premium Collection ProQuest Computer Science Collection Computer Science Database Civil Engineering Abstracts ProQuest Engineering Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional Computing Database Research Library Science Database Engineering Database Research Library (Corporate) Advanced Technologies & Aerospace Database ProQuest Advanced Technologies & Aerospace Collection Research Library China ProQuest Central Premium ProQuest One Academic (New) ProQuest One Academic Middle East (New) ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Applied & Life Sciences ProQuest One Academic ProQuest One Academic UKI Edition ProQuest Central China Engineering collection ProQuest Central Basic |
| DatabaseTitle | CrossRef Research Library Prep Computer Science Database ProQuest Central Student Technology Collection Technology Research Database Computer and Information Systems Abstracts – Academic ProQuest One Academic Middle East (New) Mechanical & Transportation Engineering Abstracts ProQuest Advanced Technologies & Aerospace Collection ProQuest Central Essentials ProQuest Computer Science Collection Computer and Information Systems Abstracts ProQuest Central (Alumni Edition) SciTech Premium Collection ProQuest One Community College Research Library (Alumni Edition) ProQuest Pharma Collection ProQuest Central China ProQuest Central ProQuest One Applied & Life Sciences ProQuest Engineering Collection ProQuest Central Korea ProQuest Research Library Research Library China ProQuest Central (New) Advanced Technologies Database with Aerospace Engineering Collection Advanced Technologies & Aerospace Collection Civil Engineering Abstracts ProQuest Computing Engineering Database ProQuest Science Journals (Alumni Edition) ProQuest Central Basic ProQuest Science Journals ProQuest Computing (Alumni Edition) ProQuest One Academic Eastern Edition ProQuest Technology Collection ProQuest SciTech Collection Computer and Information Systems Abstracts Professional Advanced Technologies & Aerospace Database ProQuest One Academic UKI Edition Materials Science & Engineering Collection Engineering Research Database ProQuest One Academic ProQuest One Academic (New) ProQuest Central (Alumni) |
| DatabaseTitleList | Civil Engineering Abstracts Research Library Prep |
| Database_xml | – sequence: 1 dbid: 8FG name: ProQuest Technology Collection url: https://search.proquest.com/technologycollection1 sourceTypes: Aggregation Database |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics Computer Science |
| EISSN | 1432-0444 |
| EndPage | 280 |
| ExternalDocumentID | 4312387321 10_1007_s00454_016_9850_y |
| GrantInformation_xml | – fundername: Deutsche Forschungsgemeinschaft grantid: FE-340/7-2 funderid: http://dx.doi.org/10.13039/501100001659 |
| GroupedDBID | -52 -5D -5G -BR -DZ -EM -Y2 -~C -~X .4S .86 .DC 06D 0R~ 0VY 199 1N0 1SB 203 28- 29G 2J2 2JN 2JY 2KG 2KM 2LR 2P1 2VQ 2WC 2~H 30V 3V. 4.4 406 408 409 40D 40E 5GY 5QI 5VS 67Z 692 6NX 78A 88I 8AO 8FE 8FG 8FW 8G5 8TC 8UJ 95- 95. 95~ 96X AABHQ AACDK AAGAY AAHNG AAIAL AAJBT AAJKR AAKPC AANZL AARHV AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYOK AAYQN AAYTO AAYZH ABAKF ABBBX ABBXA ABDZT ABECU ABFTV ABHLI ABHQN ABJCF ABJNI ABJOX ABKCH ABKTR ABMNI ABMQK ABNWP ABQBU ABQSL ABSXP ABTEG ABTHY ABTKH ABTMW ABULA ABUWG ABWNU ABXPI ACAOD ACBXY ACDTI ACGFS ACGOD ACHSB ACHXU ACIHN ACIPV ACIWK ACKNC ACMDZ ACMLO ACOKC ACOMO ACPIV ACREN ACZOJ ADHHG ADHIR ADIMF ADINQ ADKNI ADKPE ADRFC ADTPH ADURQ ADYFF ADYOE ADZKW AEAQA AEBTG AEFIE AEFQL AEGAL AEGNC AEJHL AEJRE AEKMD AEMSY AENEX AEOHA AEPYU AESKC AETLH AEVLU AEXYK AFBBN AFEXP AFGCZ AFKRA AFLOW AFQWF AFWTZ AFYQB AFZKB AGAYW AGDGC AGGDS AGJBK AGMZJ AGQEE AGQMX AGRTI AGWIL AGWZB AGYKE AHAVH AHBYD AHKAY AHSBF AHYZX AI. AIAKS AIGIU AIIXL AILAN AITGF AJBLW AJRNO AJZVZ ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMTXH AMXSW AMYLF AMYQR AOCGG ARAPS ARCSS ARMRJ ASPBG AVWKF AXYYD AYJHY AZFZN AZQEC B-. BA0 BAPOH BBWZM BDATZ BENPR BGLVJ BGNMA BPHCQ BSONS C1A CAG CCPQU COF CS3 CSCUP DDRTE DL5 DNIVK DPUIP DU5 DWQXO EBD EBLON EBS EDO EIOEI EJD ESBYG FEDTE FERAY FFXSO FIGPU FINBP FNLPD FRRFC FSGXE FWDCC GGCAI GGRSB GJIRD GNUQQ GNWQR GQ6 GQ7 GQ8 GUQSH GXS H13 HCIFZ HF~ HG5 HG6 HMJXF HQYDN HRMNR HVGLF HZ~ I-F I09 IHE IJ- IKXTQ ITM IWAJR IXC IZIGR IZQ I~X I~Z J-C J0Z JBSCW JCJTX JZLTJ K6V K7- KDC KOV KOW KQ8 L6V LAS LLZTM LO0 M0N M2O M2P M4Y M7S MA- MQGED N2Q N9A NB0 NDZJH NPVJJ NQJWS NU0 O9- O93 O9G O9I O9J OAM OK1 P19 P62 P9R PADUT PF0 PKN PQQKQ PROAC PT4 PT5 PTHSS Q2X QOK QOS R4E R89 R9I REI RHV RIG RNI RNS ROL RPX RSV RYB RZK RZZ S16 S1Z S26 S27 S28 S3B SAP SCLPG SDD SDH SHX SISQX SJYHP SMT SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 T16 TN5 TSG TSK TSV TUC TUS U2A UG4 UOJIU UTJUX UZXMN VC2 VFIZW VH1 W23 W48 WK8 YIN YLTOR Z45 Z7X Z83 Z88 Z8R Z8W Z92 ZMTXR ZWQNP ~EX AAPKM AAYXX ABBRH ABDBE ABFSG ACMFV ACSTC ADHKG AEZWR AFDZB AFHIU AFOHR AGQPQ AHPBZ AHWEU AIXLP AMVHM ATHPR AYFIA CITATION PHGZM PHGZT 7SC 7TB 7XB 8AL 8FD 8FK ABRTQ FR3 JQ2 KR7 L7M L~C L~D MBDVC PKEHL PQEST PQGLB PQUKI PRINS Q9U PUEGO |
| ID | FETCH-LOGICAL-c349t-8a58ac7d5b8a8121783df4731b606c1229de5918d9b8b3ffc4a2f2382a66d8023 |
| IEDL.DBID | AGYKE |
| ISSN | 0179-5376 |
| IngestDate | Thu Sep 04 21:06:14 EDT 2025 Sat Aug 23 14:55:55 EDT 2025 Tue Jul 01 01:43:57 EDT 2025 Thu Apr 24 23:03:13 EDT 2025 Fri Feb 21 02:43:17 EST 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 2 |
| Keywords | 05C10 Discrete harmonic functions Triangles Planar graphs 05C62 Straight line drawings 68R10 Contact family of pseudosegments |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c349t-8a58ac7d5b8a8121783df4731b606c1229de5918d9b8b3ffc4a2f2382a66d8023 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23 |
| ORCID | 0000-0003-2702-5964 |
| PQID | 1865258021 |
| PQPubID | 31658 |
| PageCount | 24 |
| ParticipantIDs | proquest_miscellaneous_1884122661 proquest_journals_1865258021 crossref_citationtrail_10_1007_s00454_016_9850_y crossref_primary_10_1007_s00454_016_9850_y springer_journals_10_1007_s00454_016_9850_y |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | 20170300 2017-3-00 20170301 |
| PublicationDateYYYYMMDD | 2017-03-01 |
| PublicationDate_xml | – month: 3 year: 2017 text: 20170300 |
| PublicationDecade | 2010 |
| PublicationPlace | New York |
| PublicationPlace_xml | – name: New York |
| PublicationTitle | Discrete & computational geometry |
| PublicationTitleAbbrev | Discrete Comput Geom |
| PublicationYear | 2017 |
| Publisher | Springer US Springer Nature B.V |
| Publisher_xml | – name: Springer US – name: Springer Nature B.V |
| References | Kobourov, S.G., Mondal, D., Nishat, R.I.: Touching triangle representations for 3-connected planar graphs. In: Proceedings of the Graph Drawing. Lecture Notes in Computer Science, vol. 7704, pp. 199–210. Springer, New York (2012) KenyonRSheffieldSDimers, tilings and treesJ. Comb. Theory Ser. B200492295317209914510.1016/j.jctb.2004.07.0011055.05032 Alam, M.J., Fowler, J., Kobourov, S.G.: Outerplanar graphs with proper touching triangle representations (unpublished) FelsnerSZickfeldFOn the number of planar orientations with prescribed degreesElectron. J. Comb.20081541p24114541182.05058 Fowler, J.J.: Strongly-connected outerplanar graphs with proper touching triangle representations. In: Proceedings of the Graph Drawing. Lecture Notes in Computer Science, vol. 8242, pp. 156–161. Springer, Heidelberg (2013) MillerEPlanar graphs as minimal resolutions of trivariate monomial idealsDoc. Math.20027439019112100989.05026 HaasROrdenDRoteGSantosFServatiusBServatiusHSouvaineDLStreinuIWhiteleyWPlanar minimally rigid graphs and pseudo-triangulationsComput. Geom.2005313161213180210.1016/j.comgeo.2004.07.0031070.65014 AertsNFelsnerSStraight-line triangle representations via Schnyder labelingsJ. Graph Alg. Appl.201519467505341260710.7155/jgaa.003721323.05093 Aerts, N., Felsner, S.: Henneberg steps for triangle representations. In: Proceedings EuroComb 2013, CRM, vol. 16, pp. 503–509. Scuola Normale Superiore, Pisa (2013) de FraysseixHde MendezPORosenstiehlPOn triangle contact graphsComb. Probab. Comput.1994302233246128844210.1017/S09635483000011390807.05028 GonçalvesDLévêqueBPinlouATriangle contact representations and dualityDiscrete Comput. Geom.2012481239254291720910.1007/s00454-012-9400-11247.52003 FelsnerSConvex drawings of planar graphs and the order dimension of 3-polytopesOrder2001181937184451410.1023/A:10106047269000984.05029 Gansner, E.R., Hu, Y., Kobourov, S.G.: On touching triangle graphs. In: Proceedings of the Graph Drawing. Lecture Notes in Computer Science, vol. 6502, pp. 250–261. Springer, Heidelberg (2010) FelsnerSLattice structures from planar graphsElectron. J. Comb.200411R1524p20353091056.05039 Schnyder, W.: Embedding planar graphs on the grid. In: Proceedings of the ACM–SIAM Symposium on Discrete Algorithms, pp. 138–148 (1990) de FraysseixHde MendezPOBarycentric systems and stretchabilityDiscrete Appl. Math.200715510791095232101610.1016/j.dam.2005.12.0091117.05033 TutteWTHow to draw a graphProc. Lond. Math. Soc.19631374376715838710.1112/plms/s3-13.1.7430115.40805 FelsnerSZickfeldFSchnyder woods and orthogonal surfacesDiscrete Comput. Geom.200840103126242965010.1007/s00454-007-9027-91148.05026 FelsnerSGeodesic embeddings and planar graphsOrder200320135150203119110.1023/B:ORDE.0000009251.68514.8b1033.05028 Lovász, L.: Geometric representations of graphs (Draft version December 11, 2009). http://www.cs.elte.hu/~lovasz/geomrep.pdf SchnyderWPlanar graphs and poset dimensionOrder19895323343101038210.1007/BF003536520675.06001 S Felsner (9850_CR7) 2008; 15 9850_CR9 D Gonçalves (9850_CR13) 2012; 48 S Felsner (9850_CR5) 2003; 20 R Kenyon (9850_CR15) 2004; 92 S Felsner (9850_CR6) 2004; 11 S Felsner (9850_CR8) 2008; 40 H Fraysseix de (9850_CR10) 2007; 155 R Haas (9850_CR14) 2005; 31 9850_CR17 9850_CR16 E Miller (9850_CR18) 2002; 7 WT Tutte (9850_CR21) 1963; 13 W Schnyder (9850_CR19) 1989; 5 H Fraysseix de (9850_CR11) 1994; 3 9850_CR20 N Aerts (9850_CR2) 2015; 19 S Felsner (9850_CR4) 2001; 18 9850_CR1 9850_CR3 9850_CR12 |
| References_xml | – reference: Schnyder, W.: Embedding planar graphs on the grid. In: Proceedings of the ACM–SIAM Symposium on Discrete Algorithms, pp. 138–148 (1990) – reference: Kobourov, S.G., Mondal, D., Nishat, R.I.: Touching triangle representations for 3-connected planar graphs. In: Proceedings of the Graph Drawing. Lecture Notes in Computer Science, vol. 7704, pp. 199–210. Springer, New York (2012) – reference: de FraysseixHde MendezPOBarycentric systems and stretchabilityDiscrete Appl. Math.200715510791095232101610.1016/j.dam.2005.12.0091117.05033 – reference: de FraysseixHde MendezPORosenstiehlPOn triangle contact graphsComb. Probab. Comput.1994302233246128844210.1017/S09635483000011390807.05028 – reference: KenyonRSheffieldSDimers, tilings and treesJ. Comb. Theory Ser. B200492295317209914510.1016/j.jctb.2004.07.0011055.05032 – reference: Gansner, E.R., Hu, Y., Kobourov, S.G.: On touching triangle graphs. In: Proceedings of the Graph Drawing. Lecture Notes in Computer Science, vol. 6502, pp. 250–261. Springer, Heidelberg (2010) – reference: FelsnerSZickfeldFOn the number of planar orientations with prescribed degreesElectron. J. Comb.20081541p24114541182.05058 – reference: FelsnerSZickfeldFSchnyder woods and orthogonal surfacesDiscrete Comput. Geom.200840103126242965010.1007/s00454-007-9027-91148.05026 – reference: Aerts, N., Felsner, S.: Henneberg steps for triangle representations. In: Proceedings EuroComb 2013, CRM, vol. 16, pp. 503–509. Scuola Normale Superiore, Pisa (2013) – reference: Lovász, L.: Geometric representations of graphs (Draft version December 11, 2009). http://www.cs.elte.hu/~lovasz/geomrep.pdf – reference: MillerEPlanar graphs as minimal resolutions of trivariate monomial idealsDoc. Math.20027439019112100989.05026 – reference: SchnyderWPlanar graphs and poset dimensionOrder19895323343101038210.1007/BF003536520675.06001 – reference: FelsnerSLattice structures from planar graphsElectron. J. Comb.200411R1524p20353091056.05039 – reference: GonçalvesDLévêqueBPinlouATriangle contact representations and dualityDiscrete Comput. Geom.2012481239254291720910.1007/s00454-012-9400-11247.52003 – reference: AertsNFelsnerSStraight-line triangle representations via Schnyder labelingsJ. Graph Alg. Appl.201519467505341260710.7155/jgaa.003721323.05093 – reference: HaasROrdenDRoteGSantosFServatiusBServatiusHSouvaineDLStreinuIWhiteleyWPlanar minimally rigid graphs and pseudo-triangulationsComput. Geom.2005313161213180210.1016/j.comgeo.2004.07.0031070.65014 – reference: TutteWTHow to draw a graphProc. Lond. Math. Soc.19631374376715838710.1112/plms/s3-13.1.7430115.40805 – reference: FelsnerSGeodesic embeddings and planar graphsOrder200320135150203119110.1023/B:ORDE.0000009251.68514.8b1033.05028 – reference: Alam, M.J., Fowler, J., Kobourov, S.G.: Outerplanar graphs with proper touching triangle representations (unpublished) – reference: FelsnerSConvex drawings of planar graphs and the order dimension of 3-polytopesOrder2001181937184451410.1023/A:10106047269000984.05029 – reference: Fowler, J.J.: Strongly-connected outerplanar graphs with proper touching triangle representations. In: Proceedings of the Graph Drawing. Lecture Notes in Computer Science, vol. 8242, pp. 156–161. Springer, Heidelberg (2013) – ident: 9850_CR9 doi: 10.1007/978-3-319-03841-4_14 – volume: 31 start-page: 31 year: 2005 ident: 9850_CR14 publication-title: Comput. Geom. doi: 10.1016/j.comgeo.2004.07.003 – ident: 9850_CR3 – volume: 20 start-page: 135 year: 2003 ident: 9850_CR5 publication-title: Order doi: 10.1023/B:ORDE.0000009251.68514.8b – volume: 18 start-page: 19 year: 2001 ident: 9850_CR4 publication-title: Order doi: 10.1023/A:1010604726900 – ident: 9850_CR16 doi: 10.1007/978-3-642-36763-2_18 – volume: 7 start-page: 43 year: 2002 ident: 9850_CR18 publication-title: Doc. Math. doi: 10.4171/dm/117 – ident: 9850_CR1 doi: 10.1007/978-88-7642-475-5_80 – ident: 9850_CR12 doi: 10.1007/978-3-642-18469-7_23 – volume: 155 start-page: 1079 year: 2007 ident: 9850_CR10 publication-title: Discrete Appl. Math. doi: 10.1016/j.dam.2005.12.009 – volume: 3 start-page: 233 issue: 02 year: 1994 ident: 9850_CR11 publication-title: Comb. Probab. Comput. doi: 10.1017/S0963548300001139 – ident: 9850_CR17 – volume: 48 start-page: 239 issue: 1 year: 2012 ident: 9850_CR13 publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-012-9400-1 – volume: 5 start-page: 323 year: 1989 ident: 9850_CR19 publication-title: Order doi: 10.1007/BF00353652 – volume: 19 start-page: 467 year: 2015 ident: 9850_CR2 publication-title: J. Graph Alg. Appl. doi: 10.7155/jgaa.00372 – ident: 9850_CR20 – volume: 15 start-page: 41p year: 2008 ident: 9850_CR7 publication-title: Electron. J. Comb. – volume: 92 start-page: 295 year: 2004 ident: 9850_CR15 publication-title: J. Comb. Theory Ser. B doi: 10.1016/j.jctb.2004.07.001 – volume: 11 start-page: 24p issue: R15 year: 2004 ident: 9850_CR6 publication-title: Electron. J. Comb. – volume: 13 start-page: 743 year: 1963 ident: 9850_CR21 publication-title: Proc. Lond. Math. Soc. doi: 10.1112/plms/s3-13.1.743 – volume: 40 start-page: 103 year: 2008 ident: 9850_CR8 publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-007-9027-9 |
| SSID | ssj0004908 |
| Score | 2.1350656 |
| Snippet | A straight line triangle representation (SLTR) of a planar graph is a straight line drawing such that all the faces including the outer face have triangular... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) A straight line triangle representation (SLTR) of a planar graph is a straight line... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).A straight line triangle representation (SLTR) of a planar graph is a straight line... |
| SourceID | proquest crossref springer |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 257 |
| SubjectTerms | Combinatorics Computational Mathematics and Numerical Analysis Contact Graphical representations Graphs Harmonic analysis Harmonic functions Mathematics Mathematics and Statistics Representations Straight lines Texts Triangles |
| SummonAdditionalLinks | – databaseName: ProQuest Technology Collection dbid: 8FG link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV07T8MwED5BWWDgUUCEl4KEGEAWediJPSGEKBVSGVArdYsc22Wp0kLaof8eX-qkgETnOLFzPvs7--6-A7jGsLU0ZpzkRgtCuTAkp9oQi7QsZjQ3MsF8595b0h3Q1yEbugu30oVV1ntitVHricI78vuQJyxi3ELSw_STYNUo9K66EhqbsBVGFmsxU7zzssqLFFVFOlQ6grQltVczWJKIMoy_SIjgLCCL37i0Mjb_-Ecr2Onsw66zF_3H5QQfwIYp2rBX12Lw3dJsw06v4V8tD-EGSWfx2O3bs6bx-1bLio-x8d-ruFeXblSURzDoPPefusRVRCAqpmJGuGRcqlSznEuLzGHKYz2iaRzm9hyiwigS2jARci1ynsejkaIyGllQjmSSaKR6O4ZWMSnMCfhSM5pqoaOAaqp4yi1QM2GsQZBSrXjgQVDLI1OOLhyHPs4aouNKhBmGiKEIs4UHt80r0yVXxrrG57WQM7dsymw1yR5cNY-twqMXQxZmMsc2nNo_tcP14K6enB-f-K_D0_UdnsF2hIhdhZedQ2v2NTcX1t6Y5ZeVUn0DcaHOXA priority: 102 providerName: ProQuest |
| Title | Straight Line Triangle Representations |
| URI | https://link.springer.com/article/10.1007/s00454-016-9850-y https://www.proquest.com/docview/1865258021 https://www.proquest.com/docview/1884122661 |
| Volume | 57 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV3dS8MwED90e9EHp1NxOqWC-KBE1jRpk8cp-0BxiGwwn0rTZD44OnHbw_zrvXRt_UCFPRXaa3pJ7np33N0vAGe2bC3wuCDKaEmYkIYopg1BS8s9zpSJfNvvfN_zuwN2O-TDrI97mle75ynJ9E9dNLulaHEY-vpECt4gi3Uoc4xPUBvLzc7TXeuzHVKmB9FZWSMWrSRPZv42yHdz9Olj_kiLptamXYF-zueyyOTlaj5TV_H7DwjHFSeyDVuZ9-k0l-KyA2smqUIlP9nByRS9Cpv3BZrrdBfOLYStDeIdjFyN00eZTZ7HxnlMq2iz5qVkugeDdqt_0yXZ-Qok9picERFxEcWB5kpEaOfdQHh6xALPVRjVxC6lUhsuXaGlEsobjWIW0RGaeBr5vrbAcftQSiaJOQAn0pwFWmraYJrFIhBo9rk06F4ETMeiUYNGvsxhnIGPW9bHYQGbnK5KaAvO7KqEixpcFK-8LpE3_iOu53sXZko4DV3hc8qRUbcGp8VjVB-bE4kSM5lbGpQgar2UGlzm2_VliL8-eLgS9RFsUOsOpLVrdSjN3ubmGJ2ZmTqBddHunGQijNfrVu_hEe8OaPMD-wHr8Q |
| linkProvider | Springer Nature |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1LT8MwDLbQOAAH3ojxLBJwAEW0adImB4R4ajw2ITQkbiVtMi5TB2wI7c_x24i7dAMkuHFu2qSOk8-J7c8A2xi2FodckNRoSZiQhqRMG2KRloecpUZFmO9cb0S1e3b1wB_G4KPMhcGwynJPLDZq3cnwjvwgEBGnXFhIOnp-IVg1Cr2rZQkN5Uor6MOCYswldlyb_rs9wnUPL8_sfO9QenHePK0RV2WAZCGTPSIUFyqLNU-FsmgXxCLULRaHQWpt-yygVGrDZSC0TEUatloZU7RlgY6qKNKiID6wEDDO8AKlAuMn543bu1Fmpixq4qHaEyROKf2q_oDGlGMESESk4D7pf0fGkbn7w0NbAN_FLEw7i9U7HqjYHIyZfB5mymoQntsc5mGqPmSA7S7ALtLe4sHfs6dd4zWtnudPbePdFZG3LuEp7y7C_b9IawkqeSc3y-ApzVmspaY-0ywTsbCmApfGmiQx05nwq-CX8kgyR1iOQ28nQ6rlQoQJBqmhCJN-FfaGrzwP2Dr-arxWCjlxC7ebjNSsClvDx3bJoR9F5abzhm0Es39qh1uF_XJyvnzitw5X_u5wEyZqzfpNcnPZuF6FSYr2QxHstgaV3uubWbfWTy_dcCrmweN_a_UnC_ARAA |
| linkToPdf | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1RT9swED4hkCb2AAw2UShbJo09bLJIHDuxHxAagwoGVAiBxFtwYocXlAJthfoX-VXcpU4LSOON5zixcz77O_vuvgP4QWFraSwVy53VTCjtWC6sY4i0MpYidyahfOeTbnJwIf5dyssZeGxyYSisstkT643a9gq6I9-KVCK5VAhJW6UPizjd6-zc3jGqIEWe1qachvFlFux2TTfmkzyO3OgBj3P97cM9nPtNzjv7538PmK84wIpY6AFTRipTpFbmyiDyRamKbSnSOMrRzi8izrV1UkfK6lzlcVkWwvASQY-bJLGqJkFAOJhLEfXxIDi3u989PZtmaeq6Ph4tAUYkKo2PNRxTmkqKBkmYVjJko5coOTV9X3lraxDsLMGCt16DP2N1-wQzrlqGxaYyROA3imX4eDJhg-2vwE-iwKVLgABPvi44R52vrm9ccFZH4frkp6r_GS7eRVpfYLbqVW4VAmOlSK22PBRWFCpVaDZI7dA8SYUtVNiCsJFHVnjychr6TTahXa5FmFHAGokwG7Xg1-SV2zFzx1uN242QM7-I-9lU5VrwffIYlx_5VEzlekNqowT-KQ63Bb-byXn2if91uPZ2h9_gA2p3dnzYPVqHeU6mRB331obZwf3QbaAhNMi_eg0L4Oq9lfoJDrEVRA |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Straight+Line+Triangle+Representations&rft.jtitle=Discrete+%26+computational+geometry&rft.au=Aerts%2C+Nieke&rft.au=Felsner%2C+Stefan&rft.date=2017-03-01&rft.issn=0179-5376&rft.eissn=1432-0444&rft.volume=57&rft.issue=2&rft.spage=257&rft.epage=280&rft_id=info:doi/10.1007%2Fs00454-016-9850-y&rft.externalDBID=n%2Fa&rft.externalDocID=10_1007_s00454_016_9850_y |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0179-5376&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0179-5376&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0179-5376&client=summon |