46:
20:
31:
233:
or edge separators, subsets of few edges whose removal disconnects the given graph into two subgraphs with approximately equal numbers of vertices. After finding an approximate cut, their algorithm arranges the two subgraphs on each side of the cut recursively, without considering the additional
143:
or social network visualization, is its neutrality: by placing all vertices at equal distances from each other and from the center of the drawing, none is given a privileged position, countering the tendency of viewers to perceive more centrally located nodes as being more important.
412:
277:
167:
The visual distinction between the inside and the outside of the vertex circle in a circular layout may be used to separate two different styles of edge drawing. For instance, a circular drawing algorithm of
540:
120:
A circular layout may be used on its own for an entire graph drawing, but it also may be used as the layout for smaller clusters of vertices within a larger graph drawing, such as its
586:
495:
452:
432:
272:
252:
215:
of the graph before combining the solutions, as these components may be drawn so that they do not interact. In general, minimizing the number of crossings is
634:(the maximum number of edges that connects one arc of the circle to the opposite arc) have also been considered, but many of these problems are NP-complete.
630:
Along with crossings, circular versions of problems of optimizing the lengths of edges in a circular layout, the angular resolution of the crossings, or the
1318:
Iragne, Florian; Nikolski, Macha; Mathieu, Bertrand; Auber, David; Sherman, David (2005), "ProViz: protein interaction visualization and exploration",
1394:
1295:
1214:
1060:
Graph-Theoretic
Concepts in Computer Science: 30th International Workshop, WG 2004, Bad Honnef, Germany, June 21-23, 2004, Revised Papers
234:
crossings formed by the edges that cross the cut. They prove that the number of crossings occurring in the resulting layout, on a graph
1507:
Algorithm
Engineering and Experimentation: International Workshop ALENEX'99, Baltimore, MD, USA, January 15–16, 1999, Selected Papers
1467:
Graph-Theoretic
Concepts in Computer Science: 20th International Workshop, WG '94, Herrsching, Germany, June 16–18, 1994, Proceedings
643:
595:
Heuristic methods for reducing the crossing complexity have also been devised, based e.g. on a careful vertex insertion order and on
187:
of one of these arcs with the vertex circle is the same at both ends of the arc, a property that simplifies the optimization of the
1545:
Biological and
Medical Data Analysis: 5th International Symposium, ISBMDA 2004, Barcelona, Spain, November 18-19, 2004, Proceedings
1524:
1418:
1255:
1185:
1562:
1392:
Masuda, S.; Kashiwabara, T.; Nakajima, K.; Fujisawa, T. (1987), "On the NP-completeness of a computer network layout problem",
497:, but this was later found to have an erroneous proof. Instead, the best approximation known for the balanced cut problem has
407:{\displaystyle O{\Bigl (}{\bigl (}\rho \log n{\bigr )}^{2}\cdot {\bigl (}C+\sum _{v\in V(G)}\deg(v)^{2}{\bigr )}{\Bigr )},}
109:, a circular layout allows the cycle to be depicted as the circle, and in this way circular layouts form the basis of the
1121:
Algorithms and
Computation: 7th International Workshop, WALCOM 2013, Kharagpur, India, February 14-16, 2013, Proceedings
1419:
Graph
Drawing: 18th International Symposium, GD 2010, Konstanz, Germany, September 21-24, 2010, Revised Selected Papers
188:
1543:
Symeonidis, Alkiviadis; Tollis, Ioannis G. (2004), "Visualization of biological information with circular drawings",
1465:
Shahrokhi, Farhad; Sýkora, Ondrej; Székely, László A.; Vrt'o, Imrich (1995), "Book embeddings and crossing numbers",
1455:
1064:
1525:
Graph
Drawing: 7th International Symposium, GD'99, Ĺ tiĹ™Ăn Castle, Czech Republic, September 15–19, 1999, Proceedings
1277:
Proceedings of the
Workshop on Information Technologies – Applications and Theory (ITAT), Slovakia, September 15-19
156:
of the circle, as circular arcs (possibly perpendicular to the vertex circle, so that the edges model lines of the
454:
is the approximation ratio of the balanced cut algorithm used by this layout method. Their work cites a paper by
1186:
Graph
Drawing: Symposium on Graph Drawing, GD '96, Berkeley, California, USA, September 18–20, 1996, Proceedings
1256:
Graph
Drawing: 14th International Symposium, GD 2006, Karlsruhe, Germany, September 18-20, 2006, Revised Papers
500:
204:
172:
uses edge bundling within the circle, together with some edges that are not bundled, drawn outside the circle.
133:
50:
607:
of crossings is within a factor of three of the maximum number of crossings among all possible layouts.
596:
184:
1183:
Doğrusöz, Uğur; Madden, Brendan; Madden, Patrick (1997), "Circular layout in the Graph Layout toolkit",
1293:(2007), "Effects of sociogram drawing conventions and edge crossings in social network visualization",
1136:
Doğrusöz, Uğur; Belviranli, M.; Dilek, A. (2012), "CiSE: A circular spring embedder layout algorithm",
1080:
Becker, Moritz Y.; Rojas, Isabel (2001), "A graph layout algorithm for drawing metabolic pathways",
599:. A circular layout may also be used to maximize the number of crossings. In particular, choosing a
1320:
549:
615:
612:
226:
39:
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1619:
1009:
157:
1445:
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212:
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when all edges are drawn inside the vertex circle. This number of crossings is zero only for
121:
66:
1522:
Six, Janet M.; Tollis, Ioannis G. (1999b), "A framework for circular drawings of networks",
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8:
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237:
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Six, Janet M.; Tollis, Ioannis G. (1999a), "Circular drawings of biconnected graphs",
1094:
1451:
1339:
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for the vertices causes each possible crossing to occur with probability 1/3, so the
208:
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102:
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1153:
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153:
90:
1212:; Kobourov, Stephen G.; Nöllenburg, Martin (2012), "Lombardi drawings of graphs",
24:
1589:
1552:
1547:, Lecture Notes in Computer Science, vol. 3337, Springer, pp. 468–478,
1528:, Lecture Notes in Computer Science, vol. 1731, Springer, pp. 107–116,
1428:
1422:, Lecture Notes in Computer Science, vol. 6502, Springer, pp. 397–399,
1265:
1259:, Lecture Notes in Computer Science, vol. 4372, Springer, pp. 386–398,
1128:
1123:, Lecture Notes in Computer Science, vol. 7748, Springer, pp. 298–309,
1072:
1034:
608:
78:
1469:, Lecture Notes in Computer Science, vol. 903, Springer, pp. 256–268,
1189:, Lecture Notes in Computer Science, vol. 1190, Springer, pp. 92–100,
1509:, Lecture Notes in Computer Science, vol. 1619, Springer, pp. 57–73,
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1005:
604:
459:
180:
129:
1585:
1384:
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1116:
1057:(2005), "Crossing reduction in circular layouts", in van Leeuwen, Jan (ed.),
1054:
997:
176:
140:
58:
35:
1514:
1352:
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1343:
1167:
1103:
873:
666:
653:
652:, a puzzle in which a player must move vertices to untangle a drawing of a
110:
98:
94:
70:
1560:
Verbitsky, Oleg (2008), "On the obfuscation complexity of planar graphs",
1149:
139:
One advantage of a circular layout in some of these applications, such as
1409:
1290:
1112:
216:
200:
132:. If multiple vertex circles are used in this way, other methods such as
114:
45:
1395:
Proceedings of the IEEE International Symposium on Circuits and Systems
1309:
1275:
1237:
1158:
1001:
211:. For other graphs, it may be optimized or reduced separately for each
1416:; Huang, Weidong (2011), "Large crossing angles in circular layouts",
649:
455:
631:
19:
1576:
1274:
He, H.; Sýkora, Ondrej (2004), "New circular drawing algorithms",
1228:
588:
on graphs that have a large number of crossings relative to their
1391:
867:
1464:
879:
222:
74:
1203:
1119:(2013), "Circular graph drawings with large crossing angles",
812:
1010:"Expander flows, geometric embeddings and graph partitioning"
897:
30:
128:
in a gene interaction graph, or natural subgroups within a
125:
1487:"Cut problems and their application to divide-and-conquer"
1317:
773:
77:, often evenly spaced so that they form the vertices of a
1138:
IEEE Transactions on Visualization and Computer Graphics
646:, a closely related concept in information visualization
203:
of the vertices of a circular layout that minimizes the
1135:
1110:
967:
761:
199:
Several authors have studied the problem of finding a
49:
Incremental construction of a circular layout for the
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503:
468:
440:
420:
280:
260:
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1407:
1182:
963:
919:
721:
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89:Circular layouts are a good fit for communications
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406:
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246:
1436:
709:
396:
286:
1611:
1542:
996:
903:
737:
1444:(2013), "2.3.2 Cubic graphs and LCF notation",
1371:Mäkinen, Erkki (1988), "On circular layouts",
1284:
1067:, vol. 3353, Springer, pp. 332–343,
785:
625:
179:, with edges drawn both inside and outside as
1498:Approximation Algorithms for NP-hard Problems
1373:International Journal of Computer Mathematics
389:
329:
313:
293:
1296:Journal of Graph Algorithms and Applications
1252:
1215:Journal of Graph Algorithms and Applications
959:
827:
656:, starting from a randomized circular layout
169:
152:The edges of the drawing may be depicted as
1401:
1079:
1052:
931:
855:
843:
808:
806:
697:
542:, giving this circular layout algorithm an
1521:
1504:
1360:Release 1.0: Esther Dyson's Monthly Report
923:
839:
797:
725:
1575:
1559:
1533:
1447:Configurations from a Graphical Viewpoint
1427:
1333:
1308:
1264:
1253:Gansner, Emden R.; Koren, Yehuda (2007),
1227:
1194:
1157:
1093:
943:
644:Chord diagram (information visualization)
535:{\displaystyle \rho =O({\sqrt {\log n}})}
1273:
927:
803:
44:
29:
18:
1370:
979:
955:
915:
823:
821:
762:Doğrusöz, Belviranli & Dilek (2012)
434:is the optimal number of crossings and
16:Graph drawing with vertices on a circle
1612:
1481:
1111:Dehkordi, Hooman Reisi; Nguyen, Quan;
891:
194:
65:is a style of drawing that places the
1350:
749:
136:may be used to arrange the clusters.
920:Doğrusöz, Madden & Madden (1997)
818:
722:Doğrusöz, Madden & Madden (1997)
686:Doğrusöz, Madden & Madden (1997)
13:
1500:, PWS Publishing, pp. 192–235
667:Circular layout engine of Graphviz
14:
1636:
1065:Lecture Notes in Computer Science
660:
904:Arora, Rao & Vazirani (2009)
1095:10.1093/bioinformatics/17.5.461
973:
949:
937:
909:
861:
849:
833:
791:
710:Pisanski & Servatius (2013)
164:), or as other types of curve.
101:, and for the cyclic parts of
84:
786:Huang, Hong & Eades (2007)
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767:
755:
743:
738:Symeonidis & Tollis (2004)
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354:
1:
1335:10.1093/bioinformatics/bth494
989:
581:{\displaystyle O(\log ^{3}n)}
147:
1563:Theoretical Computer Science
1553:10.1007/978-3-540-30547-7_47
1429:10.1007/978-3-642-18469-7_40
1353:"Visualizing human networks"
1266:10.1007/978-3-540-70904-6_37
1129:10.1007/978-3-642-36065-7_28
1073:10.1007/978-3-540-30559-0_28
134:force-directed graph drawing
7:
637:
626:Other optimization criteria
53:of social network formation
10:
1641:
960:Gansner & Koren (2007)
828:Gansner & Koren (2007)
490:{\displaystyle \rho =O(1)}
170:Gansner & Koren (2007)
105:. For graphs with a known
1586:10.1016/j.tcs.2008.02.032
1402:Baur & Brandes (2005)
1385:10.1080/00207168808803629
932:Baur & Brandes (2005)
856:Baur & Brandes (2005)
844:Baur & Brandes (2005)
698:Becker & Rojas (2001)
1535:10.1007/3-540-46648-7_11
1475:10.1007/3-540-59071-4_53
1450:, Springer, p. 32,
1196:10.1007/3-540-62495-3_40
924:Six & Tollis (1999a)
840:Six & Tollis (1999a)
798:Six & Tollis (1999a)
726:Six & Tollis (1999b)
672:
205:number of edge crossings
175:For circular layouts of
1515:10.1007/3-540-48518-X_4
1031:10.1145/1502793.1502794
880:Shahrokhi et al. (1995)
616:approximation algorithm
462:from 1994 that claimed
227:approximation algorithm
223:Shahrokhi et al. (1995)
40:border gateway protocol
23:Circular layout of the
1351:Krebs, Valdis (1996),
1204:Duncan, Christian A.;
968:Dehkordi et al. (2013)
928:He & Sýkora (2004)
582:
536:
491:
448:
428:
408:
268:
248:
122:biconnected components
54:
42:
27:
1150:10.1109/TVCG.2012.178
583:
537:
492:
449:
447:{\displaystyle \rho }
429:
409:
269:
249:
213:biconnected component
51:Barabási–Albert model
48:
34:Circular layout of a
33:
22:
1210:Goodrich, Michael T.
964:Nguyen et al. (2011)
868:Masuda et al. (1987)
813:Duncan et al. (2012)
774:Iragne et al. (2005)
611:this method gives a
550:
501:
466:
438:
418:
278:
258:
238:
1442:Servatius, Brigitte
620:approximation ratio
544:approximation ratio
195:Number of crossings
162:hyperbolic geometry
158:Poincaré disk model
1398:, pp. 292–295
1310:10.7155/jgaa.00152
1238:10.7155/jgaa.00251
1018:Journal of the ACM
601:random permutation
597:local optimization
578:
532:
487:
444:
424:
404:
364:
264:
244:
209:outerplanar graphs
189:angular resolution
185:angle of incidence
103:metabolic networks
91:network topologies
55:
43:
28:
1025:(2): A5:1–A5:37,
527:
427:{\displaystyle C}
340:
267:{\displaystyle n}
247:{\displaystyle G}
107:Hamiltonian cycle
1632:
1604:
1579:
1570:(1–3): 294–300,
1555:
1538:
1537:
1517:
1501:
1491:
1483:Shmoys, David B.
1477:
1460:
1432:
1431:
1399:
1387:
1366:
1357:
1346:
1337:
1313:
1312:
1285:Huang, Weidong;
1280:
1269:
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1248:
1231:
1199:
1198:
1178:
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1131:
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944:Verbitsky (2008)
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191:of the drawing.
113:for Hamiltonian
1640:
1639:
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1634:
1633:
1631:
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1629:
1610:
1609:
1608:
1494:Hochbaum, Dorit
1489:
1458:
1438:Pisanski, TomaĹľ
1355:
1206:Eppstein, David
1053:Baur, Michael;
1012:
1006:Vazirani, Umesh
992:
987:
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978:
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942:
938:
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605:expected number
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419:
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197:
150:
87:
79:regular polygon
63:circular layout
17:
12:
11:
5:
1638:
1628:
1627:
1622:
1607:
1606:
1557:
1540:
1519:
1502:
1479:
1462:
1456:
1434:
1414:Hong, Seok-Hee
1408:Nguyen, Quan;
1405:
1400:. As cited by
1389:
1368:
1348:
1328:(2): 272–274,
1321:Bioinformatics
1315:
1303:(2): 397–429,
1287:Hong, Seok-Hee
1282:
1271:
1250:
1201:
1180:
1144:(6): 953–966,
1133:
1117:Hong, Seok-Hee
1108:
1088:(5): 461–467,
1082:Bioinformatics
1077:
1055:Brandes, Ulrik
1050:
998:Arora, Sanjeev
993:
991:
988:
985:
984:
980:Mäkinen (1988)
972:
956:Mäkinen (1988)
948:
936:
916:Mäkinen (1988)
908:
896:
884:
872:
860:
848:
832:
817:
802:
790:
778:
766:
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674:
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661:External links
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624:
590:vertex degrees
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460:Shing-Tung Yau
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196:
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177:regular graphs
149:
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141:bioinformatics
130:social network
124:, clusters of
86:
83:
15:
9:
6:
4:
3:
2:
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1626:
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1620:Graph drawing
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1457:9780817683641
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1222:(1): 85–108,
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892:Shmoys (1997)
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610:
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228:
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36:state diagram
32:
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25:Chvátal graph
21:
1567:
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1410:Eades, Peter
1393:
1379:(1): 29–37,
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1291:Eades, Peter
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654:planar graph
629:
594:
221:
198:
174:
166:
151:
138:
119:
115:cubic graphs
111:LCF notation
88:
85:Applications
62:
56:
1159:11693/21006
1002:Rao, Satish
217:NP-complete
201:permutation
1614:Categories
990:References
148:Edge style
1577:0705.3748
1229:1009.0579
650:Planarity
570:
522:
505:ρ
470:ρ
456:Fan Chung
442:ρ
369:
349:∈
342:∑
325:⋅
305:
299:ρ
229:based on
1485:(1997),
1344:15347570
1176:14365664
1168:23559509
1104:11331241
1047:52151977
1008:(2009),
638:See also
632:cutwidth
93:such as
67:vertices
38:for the
1625:Circles
1602:5948167
1594:2412266
1496:(ed.),
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622:three.
1600:
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414:where
183:, the
154:chords
75:circle
1598:S2CID
1572:arXiv
1492:, in
1490:(PDF)
1356:(PDF)
1242:S2CID
1224:arXiv
1172:S2CID
1043:S2CID
1013:(PDF)
673:Notes
618:with
254:with
126:genes
73:on a
71:graph
69:of a
1452:ISBN
1364:2–96
1340:PMID
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1100:PMID
458:and
95:star
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