Proximity Drawing, 117 1-Outerplanar Embedded Graph, 263 2-Claw, 266

Index

(1 + ε)-EMST drawing, 139 t-track layout, 460 (h, k)-proximity drawing, 117 relative position, 711 1-outerplanar embedded graph, 263 Monotone3Sat, 700 2-claw, 266 MonotonePlanar3Sat, 700 Co(G)-component, 324 Partition, 704 P GE, 328 Planar3Sat, 713 P GW , 328 1-tough, 126 Gd, 321 3-cycle, 61 Kn, 43, 44, 47, 48 3D drawing, 665, 683 Km,n, 46, 48 3D embedding, 606, 612 Kn,m, 43 3D grid drawing, 457 NF 3-graphs, 138 3D hyperbolic space, 793 PE, 324 3D orthogonal drawing, 466 PN , 324 3D polyline drawing, 465 PS, 324 PW , 324 Qcc(C), 327 Qc(C), 327 ncc(C), 327 ∆, 318 ∆-drawing, 675 ABACUS, 545, 563 Θ, 321 access control, 668 Θ(G), 49 accumulation tree, 435 α-complexes, 143 Ackermann function, 68 β-drawings, 119 acyclic, 224 γ-drawings, 120 acyclic subgraph, 547 γ-regions, 120 adaptive tree drawing, 184 bw(G), 49 aesthetic criteria, 122 crs(G), 57 aesthetics, 157 cr(G), 45 AGD, 544, 563 cr1(G), 45 AJAX, 694 crt(G), 46 algebraic pricing, 63 cw(G), 49 algorithm ocr(G), 45 Auslander-Parter, 12–14 pcr(G), 45 Boyer-Myrvold, 27–31 sk(G), 48 Coffman-Graham, 421 ε1-shrunk disk of D, 140 de Fraysseix-Ossona de Mendez-Rosenstiehl, ε2-expanded disk of D, 140 15–17 d-linear, 712 Hopcroft-Tarjan, 14–15 k-connected, 224 k-means, 727 k-localized Delaunay triangulations, 143 Lempel-Even-Cederbaum, 21–24 k-outerplanar, 263 Rectangular-Draw, 332 k-outerplanar embedded graph, 263 Shih-Hsu, 24–27 k-outerplanar embedding, 263 algorithm animation, 738, 744 k-planar graphs, 225 algorithm simulation, 740 k-relative neighborhood drawing, 118 almost planar graph, 77 nc(C), 327 amino acid, 625 st-graph, 225 angular resolution, 223, 225, 712 842 INDEX animation, 627 book embedding, 251 apex graph, 77 Border Gateway Protocol, 665 approximation algorithm, 76 botanical tree, 184 approximation factor, 68 boundary face, 328 architectural floorplanning, 319 boundary path, 328 area, 157 bounding box, 458 area of a grid drawing, 225 box-drawings, 470 Artificial graphs, 73 box-rectangular drawing, 319, 337 aspect ratio, 157 branch-and-bound tree, 62 assignment heuristic, 435 branch-and-cut, 55, 66, 437 augmentation, 547, 606 branch-and-cut approach, 62 biconnected, 547 branching variable, 62 fixed embedding, 547 branding, 682 planar biconnected, 547 bridge, 3, 714 Automatic label placement, 489 brin, 601 automorphism, 89 bush, 22 Autonomous Systems, 665 bush form, 22 autonomous systems, 779 averaging, 435 c-connected, 561 Axial Geometric Automorphism Problem, C-node, 67 96 c-planarity, 561 canonical order, 550 B-node, 67 canonical ordering, 199–202, 240, 260 bad cycle, 324 canonical ordering , 227 barycenter, 435, 747 capacity, 231 barycenter heuristic, 555 caterpillar, 266 barycenter method, 197 CCMP, 68 barycentric representation, 216 cellular compartment, 624 barycentric representations, 383 centrality, 814 BC-tree, 3, 67, 68, 547 characteristic matrix, 635 dynamic, 548 checkers, 144 bend, 156, 248 chordal, 369 bend minimization, 238 chromatic number, 457, 459, 461 BGP, 665, 779, 794 circle, 631 biconnected, 197, 224, 288 circle layout, 640 biconnected component, 3 circle packing, 368 biconnected components, 197, 547 circular crossing minimization problem, 54 binary tree, 163 circular drawing, 285 biological network, 621 cladistic methods, 635 bipartite, 54 cladogram, 640 bipartite drawing, 79 class-cover catch digraphs, 143 bipartite crossing minimization problem, 54 clockwise leg, 327 bipartite crossing number problem, 50 closed proximity drawing, 117 bipartite drawing, 50 closed strip drawings, 119 bisection width, 49, 77 cloud computing, 673, 693 block, 3 cluster tree, 561 block tree, 547 clustered drawing, 656, 816 block-cutvertex tree, 3, 67 clustered graph, 31, 561 block-nesting depth, 548 clustered planarity, 31 blocks, 197 cohesion, 815 INDEX 843

COIN-OR, 545, 563 crossing minimization problem, 43, 63 colored simultaneous embedding, 365 fixed linear, 46 column generation, 62 crossing number, 43, 45, 288 combinatorial column generation, 63 crossing number matrix, 434 combinatorial map, 601 crossing number problem, 50, 563 compaction, 552, 580 crossing reduction, 288, 292 comparability graph, 369 crossing shadow, 61 compartment, 624 crossing-reduction, 747 complete bipartite graph, 43, 49 curve complete graph, 49 Bézier,718 complex biological network, 623 cubic, 701 component Bézier biconnected, 3 cubic, 725 connected, 3 Catmull–Rom spline, 709 triconnected, 3 curve embedding, 260 compound, 621 curvogram, 640 compound graph, 562 cutvertex, 3, 67, 71 compressed-scanning, 674 cutwidth, 49 computer games, 682 cycle computer network, 683 attached, 326 computer networks, 763 clockwise, 327 cone trees, 162 closed, 631 connected, 197, 224 counterclockwise, 327 connected component, 3 definition, 3 connectivity, 605 fundamental, 15 constrained edge insertion, 68 length, 3 constrained visibility representation, 234 open, 631 constraint graph, 552 cycle space, 439 constraints, 55, 573, 585, 745, 753 hard, 715 DAG, 224, 675 soft, 715 dart, see brin contact representation data analytics, 682 contacts of segments, 612 data privacy, 673 contacts of T, 612 data reduction, 699 contraction data-oblivious, 674 edge, 714 decision graph, 320, 321 convex, 196 degree, 178 convex drawing, 196, 212, 220, 609 degree of a face, 236 convex faces, 550 degree of a vertex, 224 convex hull, 55 Delaunay drawing of order h, 121 coordinates assignment, 555 Delaunay triangulation, 121 counterclockwise leg, 327 dendrogram, 639 country map, 663 depth first search, 10, 288 cover tree, 200 highpoint, 11 critical cycle, 327 index, 11 crossing, 45, 460 lowpoint, 11 crossing minimization, 548 tree, 11 2-layer, 555 depth of a graph, 8 k-layer, 555 Depth-First Search, see DFS upward, 557, 564 design patterns, 687 844 INDEX destination, 224 self-loop, 2 details on demand, 687 subdivision, 2 DFS, see depth first search, 603 transitive, 725 DFS cotree-critical graph, 603 virtual diagram of a bush, 22 Voronoi, 727 of a skeleton, 3 digraph, 224 edge insertion, 550 directed acyclic graph, 224, 675 fixed embedding, 70, 550 directed edge, 224 general framework, 68 directed graph, 224 permutation, 69 directed local minimum spanning trees, 143 postprocessing, 69 directed path, 224 upward, 557 directed relative neighborhood graphs, 143 variable embedding, 71, 550 direction edge insertion problem, 64 octilinear, 714 education, 737 directions EIP, 64 octilinear, 712, 713, 715, 716 ELP (Edge Label Placement) problem, 497 disconnected graph, 104 email, 793 discrete curve evolution, 713 embedded planar graph, 248 distance matrix, 635 embedding, 3, 195, 225, 248 DNA, 621, 622 simultaneous, 31 document type definition, 519 embedding circle, 285 dominance drawing, 675 empty region graphs, 120 dominance drawings, 557 empty tree, 156 drawing, 2, 45 enclosing rectangle, 156 c-planar, 31 enzyme, 628 orthogonal, 712 network, 630 outerplanar, 2 Euler inequalities, 66 planar, 2 Euler tour, 675 drawing style Euler’s formula, 47, 197 curvilinear, 718 evolution, 623 drawing convention, 156 evolutionary tree, 636 drawing style executive dashboards, 684 octilinear, 712–714 exercise systems, 740 dual edge, 196 exploration systems, 741 dual face, 195 extended dual graph, 70 dual graph, 6, 195, 256 Extensible Stylesheet Language Transfor- dual vertex, 195 mations, 534 dual-like, 319 dummy vertex, 64 face, 2 dynamic graph, 401 external, 2 dynamic graph drawing, 369, 752 outer, 2 dynamic graphs, 823 faces, 195 facet, 57 edge fan triplet, 271 adjacent, 2 fast multipole multilevel embedder, 559 bundling, 711 fast multipole multilevel method (FM3), 559 dummy, 724 feasible solution, 55 incident, 2 feedback arc set, 547 minimal, 725 file permissions, 668 INDEX 845

filled triangle, 138 co-authorship, 728 filter, 687 complete, 6 Fixed Point Free Automorphism Problem, connected, 3 96 definition, 2 fixed-parameter tractable, 54 depth, 8 fixed-parameter tractable, 434 directed, 2 flag, see brin induced by a vertex set, 2 flow, 231 induced by an edge set, 2 flow network, 231 intersection, 2 flow value, 231 near-plane, 698, 711 force directed, 97 outerplanar, 2 force-directed, 370, 383, 656, 657, 659, 666, outerplanarity, 8 744, 753 planar, 2 format conversion, 536 plane, 3, 699 frame graph, 138 simple, 2 fundamental cycle, 15 simply connected, 3 subdivision, 2 Gabriel drawing, 117 subgraph, 2 Gabriel graph, 115 triconnected, 3 Gabriel region, 117 undirected, 2 Galois lattice representation, 821 union, 2 GDToolkit, 571 width, 8 GEM algorithm, 559 graph drawing gene regulation, 623 box-rectangular drawing, 319, 337 gene regulatory rectangular drawing, 317 network, 622, 623 graph theoretic distance, 388 pathway, 623 GraphML, 517 generalization, 697 greedy insert heuristic, 555 continuous, 700 greedy insertion, 435 genus, 78 greedy switch heuristic, 555 geographic information system (GIS), 697 greedy switching, 435 geometric automorphism, 91 grid Geometric Automorphism Drawing Prob- integer, 714 lem, 95 grid drawing, 156, 197, 225 geometric automorphism group, 91 grid-variant, 559 geometric network, 697 geometric simultaneous embedding, 31 half-edge, see brin geospatial map, 684 Hamiltonian, 249 GFLP (Graphical Feature Label Placement) Hamiltonian cycle, 249 problem, 493 Hamiltonicity, 274 global proximity, 116 head vertex, 327 GMap, 727 hexagonal tree drawing, 185 gml2pic, 546 hierarchical index, 410 good cycle, 324 hierarchical layout, 744, 753 good drawing, 46 hierarchy, 411 graph higher order Delaunay triangulation, 121 k colorable, 6 hill climbing, 714 biconnected, 3 hull bipartite, 6 convex, 724 clustered, 31 hyperbolic space, 397 846 INDEX hyperbolic tree, 183, 397 definition, 492 hypercube, 49 Large Parsimony, 638 hyperedge, 76 lattice representation, 821 hypergraph, 76, 527, 820 layer-by-layer sweep, 438 hypervertex, 76 layered drawing, 184, 247, 248, 666, 670, hypothetical taxonomic units, 636 672 layered planar graph, 272 inclusion drawing, 816 layering, 633 incoming edge, 224 proper, 633 incremental planarity testing, 67, 68 layout method indegree, 224 force-directed information visualization, 687 Fruchterman-Reingold, 725 information-seeking mantra, 687 left-right partition, 15 inner angle, 320 level graph, 411 inner triangulated plane graphs, 138 life science, 621 integer linear program, 55 linear crossing minimization problem, 54 interconnection technologies, 764 linear crossing number, 46 Internet service provider, 785 linear drawing, 46 Internet structure, 765 linear ordering, 436 Internet visualization, 779 linear ordering problem, 60 intersection drawings, 122 linear program, 55 interval representation, 369 LineDrive, 699 intrusion detection, 656, 793 linked bar charts, 687 IPv6, 793 linkless, 479 isomorphism, 89 local minimum spanning trees, 143 ISP network visualization, 785 local network, 789 Lombardi drawing, 400 k-layered drawing, 248 lower bound, 236 k-radial drawing, 249 LP, 55 k-spine drawing, 249 LR-partition, 15 Kandinsky, 576 lune, 118 Kandinsky layout, 553 lune-based, 119 kissing number, 124 knowledge discovery, 687 Manhattan distance, 197 Kuratowski subdivision, 603 map, see combinatorial map Kuratowski inequalities, 66 mental, 698 Kuratowski subdivision, 57, 563 metro, 711, 712 Kuratowski’s theorem, 55 schematic, 711, 712 street, 711, 720 Label placement, 489 matched drawings, 367 label quality matching, 322 rules for the MLP problem, 504 maximum, 322 basic rules, 490 perfect, 322 LabelHints, 714 maximal planar graph, 195, 225 Labeling algorithms, 489 maximal planar subgraph, 67, 68 for the ELP problem, 497 maximum degree of a graph, 224 for the GFLP problem, 493 Maximum Likelihood, 638 for the MLP problem, 504 maximum matching, 322 for the NLP problem, 502 Maximum Parsimony, 637 Labeling Problem, 489 maximum planar subgraph, 66 INDEX 847 maximum planar subgraph problem, 48, 64, multichip module, 319 549 multidimensional scaling, 97, 396 maximum weight planar subgraph, 66 multigraph, 224 maximum weight triangulations, 143 multilevel algorithms, 559 MCM, 319 median, 435 near-planar graph, 77 median heuristic, 555 Nearest Neighbor Interchange, 638 mental map, 402, 701, 752 nearest neighbor drawing, 121 metabolic negative witness proximity drawing, 134 network, 623, 628 Neighbor-Joining, 637 path, 628 neighbors, 224 pathway, 628, 630 nested graphs, 525 reaction, 628 network simplified network, 630 biological, 621 metabolite, 628 complex biological, 623 network, 630 enzyme, 630 metaphor gene regulatory, 622, 623 metro-map, 718 metabolic, 622, 628 political map, 727 metabolite, 630 min-cost flow formulation, 724 protein-protein interaction, 622 minimal energy state, 385 signal transduction, 622, 623 minimum spanning tree, 122 simplified metabolic, 630 minimum weight drawable (for C), 122 network model, 764 minimum weight drawing of G with respect network monitoring, 656 to P , 122 network scans, 659 minimum weight triangulation, 123 network traffic, 657 minor, 457 NLP (Node Label Placement) problem, 502 minor crossing number, 76 non-planar core, 563 minor-closed, 457 non-upward drawing, 156 MIP solver nonbiconnected, 296, 297 Cplex, 716 NP-complete, 50, 55 Gurobi, 716 NP-hard, 44, 48–50 mixed integer linear program, 55 NS-path, 328 mixed-model layouts, 551 MLP (Multiple Label Placement) problem, objective function, 55 504 octopus map, 663 modified Gabriel drawing, 117 odd-crossing number, 45 modified Gabriel region, 117 OGDF, 543–564 Modified Optimal Linear Arrangement, 288 OGML, 546 module options, 545 open proximity drawing, 117 module types, 545 Open Graph Drawing Framework, 543 modules, 317, 545 Open Graph Markup Language, 546 molecular biology Open Problems Project, 699 central dogma, 621 open strip drawings, 119 moment curve M, 458 operational taxonomical units, 636 morphing, 371 optimal linear arrangement problem, 50 polygonal line, 700 optimal solution, 55 MPSP, 64 optimization multi-dimensional scaling, 727 multicriteria, 714 multi-scale method, 391 order dimension, 212 848 INDEX ordered tree, 155 permutation graphs, 369 orientation, 60 phenetic methods, 635 origin, 224 phylogenetic tree, 636 orthogonal box drawings, 456 phylogram, 639 orthogonal drawing, 156, 225, 234, 580 PIGALE, 599 orthogonal layouts, 552 geometric graph, 603 bend minimization, 552 graph, 603 orthogonal representation, 576 graph properties, 602 orthogonal shape, 236, 574 map, 601 outdegree, 224 acir, 602 outer angle, 320 cir, 602 outer vertex, 19 pbrin, 602 outerplanar, 291 vin, 602 outerplanar embedded graph, 263 supported platforms, 599 outerplanar graph, 195 topological graph, 603 outerplanarity, 9, 263 plan measure, 8 cable, 712, 724 outgoing edge , 224 planar, 195, 248 overview, 687 planar st-graph, 225 overview first, zoom and filter, then details planar augmentation problem, 547 on demand, 687 planar automorphism, 98 planar automorphism group, 98 P-node, 67 planar drawing, 156, 225 pagenumber, 251 planar drawing algorithm, 44 pages, 251 planar embedding, 225, 574 pairwise crossing number, 45 maximal external face, 548 palm tree, see depth first search, tree minimal block-nesting depth, 548 parsimony planar graph, 225, 459, 675 large, 638 planar layout, 550 maximum, 637 straight-line, 550 small, 637 planar orthogonal drawing, 238 weighted small, 637 planar polyline drawing, 239 partial Delaunay triangulations, 143 planar straight-line drawing, 193–220 partial tree, 166 planar subgraph, 44, 64, 66 partition planar subgraph polytope, 66 aligned, 16 planar subgraphs, 549 left-right, 15 planarity LR-partition, 15 clustered, 31 partition-pair, 330 constrained, 8 partitioning path, 328 simultaneous, 31 path testing, 603 addition, see algorithm, Hopcroft and upward, 9 Tarjan planarity test, 57, 67 definition, 3 planarity testing, 548, 576 length, 3 planarization, 711 PC-tree, 24 planarization approach, 44, 63, 552, 563 peer-to-peer networks, 793 clustered graphs, 561 peptide bonds, 625 planarized representation, 64 perfect matching, 322 plane dual, 6 permutation, 434 plane graph, 225 INDEX 849

dual, 6 optimal, 554 podevsnef, 576 reactant, 628 point real-time visual reports, 685 characteristic, 701 realizability problem, 57 point-drawings, 466 realizable, 58, 59 point-set embeddability, 276 realizer, 212 point-set embedding, 253, 276 realizer method, 199, 212–220 Polrec drawing, 610 rectangle of influence, 120 polygon, 196 rectangle of influence drawing, 120 polygonal line simplification, 699, 712, 713 rectangle visibility, 478 polyhedral combinatorics, 55 rectangular drawing, 317 polyline drawing, 156, 225, 248, 580 rectangular dual, 344 ports, 528 rectilinear crossing number, 45 positive witness proximity drawing, 134 rectilinear drawing, 610 PQ-tree, 68 recurrent hierarchies, 444 PQ-trees, 548 region of influence, 116 PrEd, 714 regular labeling, 321 privacy settings, 671 relative neighborhood drawing, 118 product, 628 relative neighborhood region, 118 program visualization, 741 relative position, 714 projective graph, 78 relatively closest drawing, 118 prominence, 812 relatively closest region, 118 proper layered drawings, 267 returning edge, 11 protein, 622, 625 ringed circular layout, 165, 178, 180 protein-protein interaction, 625 RNA, 622 network, 622, 625 Rome graphs, 56, 73, 563 proximity drawing checker, 144 root proximity graph, 115 of connected component, 24 proximity region, 116 route map, 699 Public Implementation of a Graph Algo- route planner, 699 rithm Library and Editor, see PI- routing protocols, 765 GALE S-node, 67 Q-node, 67 SBGN, 645 quasi-orthogonal layouts, 552 SCHEMAP, 724 quasi-static graph, 601 SeeNet3D, 726 quasi-upward, 578 semantic constraints, 247 quasi-upward planar representation, 579 sensor networks, 793 queue layout, 463 separating k-set, 196 quicksort, 435 separating edge, 197 separation pair, 3, 71 R-node, 67 separation pairs, 196 radial drawing, 260, 269, 666 separation problem, 62 radial drawings, 159, 247 separator edge, 162, 166 random field layout, 725 series-parallel graph, 675 random map, 609 shelling order, 550 rank assignment, 554 shift method, 199, 202–212 ranking shifting set, 227 Coffman-Graham, 554 sifting, 438 longest paths, 554 sifting heuristic, 555 850 INDEX signal transduction, 623 straight-line, 157 network, 622, 623 Straight-line drawing, 609 pathway, 623 straight-line drawing, 195, 225, 248 simple, 58 streaming, 823 simple drawing, 57, 58 stress majorization, 396 simulated annealing, 386, 559 strictly convex, 196 simultaneous embedding, 31, 351, 364 strictly convex drawing, 196 geometric, 31 strong visibility drawing, 136 with fixed edges, 31 sub-Hamiltonian, 249 simultaneous embedding with fixed edges, subdivision, 57 350, 357 substance, 628 simultaneous geometric embedding, 350, 352 co-, 628 simultaneous planarity, 31 main, 628 single edge insertion, 68 substance-based designs, 810 sink, 224, 231 subtree, 156 skeleton, 124, 548 subtree-separation, 158 sketch, 724 Sugiyama framework, 410 skewness, 48 Sugiyama method, 410 Small Parsimony, 637 Sugiyama’s framework, 554, 562, 564 social networks, 671, 793, 805 survivable telecommunication networks, 287 sociomatrix, 817 symmetric lens, 118 software visualization, 743 symmetry, 87, 91, 385, 480, 606, 675 source, 224, 231 space tree, 184 t-polygonal crossing number, 46 spanning tree, see tree, spanning a graph, t-polygonal drawing, 45 67, 68 tail vertex, 327 spectral analysis, 606 task-specific visualizations, 688 sphere of influence, 122 taxon, 634 sphere-of-attraction graphs, 143 taxonomical units spine, 171, 251 hypotetical taxonomic, 636 spine drawing, 251, 266, 268 operational taxonomic, 636 spine drawings, 247, 249 template region, 120 spiral, 631 tessellation representation, 368 split, 435 theorem component, 3 four-color, 728 operation, 3 thickness, 49, 473 split heuristic, 555 three-dimensional straight-line grid draw- SPQR tree, 101 ing, 457 SPQR-tree, 4, 67, 68, 547, 563, 576 timetable graph, 725 dynamic, 548 topological book embedding, 255 spring, 385 topological ordering, 225 spring embedder, 610 topology-shape-metrics approach, 552, 561, spring embedder, 385, 559, 656, 714 573 square drawings, 134 toroidal graph, 78 st-numbering, 22 track-number, 460 s-t-planar digraph, 558 trajectory, 701 star insertion, 74 transitive star triangulation, 99 closure, 421 status, 812 edge, 421 stochastic heuristic, 435 reduction, 421 INDEX 851 tree, 675 visual searching, 687 BC-tree, 3 visualization principles, 807 PC-tree, 24 VLSI, 43 PQ-tree, 24 volume, 458 spanning a graph, 2 Voronoi diagram, 121 SPQR tree, 4 Voronoi drawing, 121 tree layouts, 554 treemap, 662, 670, 672, 675 wall, 65 trees, 752 weak proximity drawings, 135 triangulated plane graphs, 138 weak visibility drawing, 136 triconnected, 197, 224 web, 793 triconnected component, 3 web applications, 694 triconnected components, 67, 197, 547, 563 Weighted Small Parsimony, 637 Tutte, 197 width Tutte drawing, 610 of a graph, 8 Tutte’s barycenter method, 559 window, 438 two-mode network, 819 wireless networks, 793 witness Delaunay drawings, 134 unit distance graph, 143 Witness proximity, 133 Unweighted Pair Group Method with Arith- Witness proximity drawings, 133 metic Mean, 636 upward, 464 X-crossing, 460 upward drawing, 76, 156, 578, 670, 675 XML, 517 upward planar representation, 557, 578 XML schema, 519 upward planar subgraphs XPath, 534 feasible, 557 XSLT, 534 upward planarity, 9 zoom, 687 upward planarity testing sT -digraphs, 549, 564 upward planarization, 557 upward representation, 578 user interface, 682 user-grouped circular drawings, 303 vertex adjacent, 2 outer, 19 virtual, 22 vertex filtration, 393 vertex insertion, 74 virtual edge of a bush, 22 of a skeleton, 3 virtual vertex, 22 visibility drawing, 610 visibility representation, 230, 234, 478, 557, 580 visual design patterns, 687 visual discovery, 686 visual presentation, 682