Straight-Line Drawing Algorithms for Hierarchical Graphs and Clustered Graphs
ABSTRACT Hierarchical graphs and clustered graphs are useful non-classical graph models for structured relational information. Hierarchical
graphs with layering structures; clustered graphs are graphs with
recursive clustering structures. Both have applications in CASE tools, software visualization and VLSI design. Drawing algorithms
graphs have been well investigated. However, the problem of planar straight-line representation has not been solved completely.
In this paper we answer the question: does every planar hierarchical graph admit a planar straight-line
hierarchical drawing? We present an algorithm that constructs
such drawings in linear time. Also, we answer a basic question for clustered
graphs, that is, does every planar clustered graph admit a planar
straight-line drawing with clusters drawn as convex polygons? We
provide a method for such drawings based on our algorithm for
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ABSTRACT: Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in VLSI de- sign, CASE tools, software visualisation and visualisation of social networks and bi- ological networks. Straight-line drawing algorithms for hierarchical graphs and clus- tered graphs have been presented in (P. Eades, Q. Feng, X. Lin and H. Nagamochi, Straight-line drawing algorithms for hierarchical graphs and clustered graphs, Algo- rithmica, 44, pp. 1-32, 2006). A straight-line drawing is called a convex drawing if every facial cycle is drawn as a convex polygon. In this paper, it is proved that every internally triconnected hierarchical plane graph with the outer facial cycle drawn as a convex polygon admits a convex drawing. We present an algorithm which constructs such a drawing. We then extend our results to convex representations of clustered planar graphs. It is proved that every internally triconnected clustered plane graph with completely connected clustering structure admits a convex drawing. We present an algorithm to construct a convex drawing of clustered planar graphs.J. Discrete Algorithms. 01/2010; 8:282-295.
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ABSTRACT: Graph visualization has been widely used to understand and present both global structural and local adjacency information in relational data sets (e.g., transportation networks, citation networks, or social networks). Graphs with dense edges, however, are difficult to visualize because fast layout and good clarity are not always easily achieved. When the number of edges is large, edge bundling can be used to improve the clarity, but in many cases, the edges could be still too cluttered to permit correct interpretation of the relations between nodes. In this paper, we present an ambiguity-free edge-bundling method especially for improving local detailed view of a complex graph. Our method makes more efficient use of display space and supports detail-on-demand viewing through an interactive interface. We demonstrate the effectiveness of our method with public coauthorship network data.IEEE transactions on visualization and computer graphics. 05/2012; 18(5):810-21.
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ABSTRACT: This paper proposes an organized generalization of Newman and Girvan's modularity measure for graph clustering. Optimized via a deterministic annealing scheme, this measure produces topologically ordered graph clusterings that lead to faithful and readable graph representations based on clustering induced graphs. Topographic graph clustering provides an alternative to more classical solutions in which a standard graph clustering method is applied to build a simpler graph that is then represented with a graph layout algorithm. A comparative study on four real world graphs ranging from 34 to 1133 vertices shows the interest of the proposed approach with respect to classical solutions and to self-organizing maps for graphs.Neurocomputing. 01/2010;
Hierarc hicalgraphs and clusteredgraphs areusefulnon?classicalgraph mo delsforstructured
relationalinformation? Hierarc hicalgraphs aregraphswith layering structures?clustered graphs
are graphs withrecursiv eclustering structures? Both have applications in CASEto ols?softw are
visualization?andVLSIdesign? Drawing algorithmsforhierarchical graphshavebeenw ellinv es?
tigated?However?theproblemofstraight?line representationhas notbeen solved completely?
Inthis paper?weanswerthe question?doesevery planarhierarchicalgraphadmit a planar
straight?linehierarchicaldra wing?We presentan algorithmthatconstructs such drawingsin
lineartime? Also?we answera basicquestion forclusteredgraphs?thatis?doeseveryplanar
clusteredgraph admitaplanar straigh t?linedrawingwith clustersdrawnasconvexpolygons?
We provideametho dforsuchdrawingsbasedon ouralgorithm forhierarchical graphs?
Keyw ords? Computationalgeometry?automaticgraphdrawing?hierarc hicalgraph?clustered
graph?straigh t?linedra wing?
AgraphG??V?E?consistsof aset Vof verticesanda setE of edges?that is?pairs ofvertices?
Graphsare commonlyused tomodelrelationsin computing?andman ysystemsformanipulating
graphsha verecen tlybeen develop ed?Examples includeCASE tools??? ??kno wledgerepresen tation
systems ????? software visualizationtools ?????and VLSI design systems?????A gr aphdrawing
algorithm readsasinputa combinatorial description ofagraph? and pro ducesas outputa visual
represen tationof thegraph? Such algorithms aimto produce drawingswhich areeasy toread and
easytoremember? Manygraph drawing algorithmsha vebeen designed?analyzed?testedand used
in visualizationsystems ?? ??
Withincreasing complexityofthe informationthatwewanttovisualize?weneed morestructure
ontopof theclassicalgraph model? Severalextendedgraph models havebeenproposed?????????
????? ??In thispaper? weconsider twosuchmo dels?
Thisworkwassupportedbyaresearch grantfromthe AustralianResearch Councilandthesubsidy fromKy oto
Universit yFoundation?An extendedabstractofthispaper was presentedat the Symp osiumonGraph Drawing
?GD?????Berk eley?California? ?????
Department ofComputerScienceandSoftw areEngineering?Universityof Newcastle?University Drive?
Callaghan? NSW????? Australia?Email?feades?qwfengg?cs?newcastle?edu?au
Scho olofComputerScience andEngineering?University of NewSouthW ales?Sydney NSW?????Australia?
DepartmentofAppliedMathematicsand Physics? Ky otoUniversity?Kyoto?????????Japan?Email?
FR in AI
Intro to CS
Figure??An exampleof a hierarchical graph?
?Hier archic al graphsaregraphswithv erticesassigned tolayers?Hierarchicalgraphsappearin
applicationswhere hierarc hicalstructures areinvolv ed????????Forexample?Figure?sho ws
aprerequisite diagramforsubjectsin aBachelor?s degree inComputer Science?an edgefrom
ato bmeansthataisaprerequisiteof b?and eachlayer representsasemesterofstudy ? In
the exampleofFigure?? thev erticesareassigned to lay ers forsemantic reasons?Inother
examples?the lay erassignmentischosen toimprovethereadabilityofthe drawing?see? for
example? ??? ??
? Clustered gr aphs aregraphs withrecursiv eclustering structureswhichapp earinman y struc?
tureddiagrams???? ????? ??? ??F orexample? Figure? shows arelationaldiagram ofsome
organizations inNew SouthW ales?an edgeb etw eenaandbindicatesajoin tpro jectbet w een
a andb? andeachcluster representsagroupof organizations?
Bothhierarc hical graphsandclustered graphs havethe pow er ofrepresenting certainadditional
structuresrequiredbyapplications? Thedrawings of hierarchical graphs andclustered graphs
shouldre?ectthese structures?andthereforem ustmeet additional constraints?
A graphG??V?E?isdra wnby specifying a locationintheplane for eachv ertexinVand
aroute ?asimpleJordancurv e?foreac hedge inE? Thedra wing isplanar if nopair ofedge
Figure??Anexample of aclustered graph?
routes cross? andthegraph isplanarifit admitsaplanardrawing?Theplanarit y property has
b eenthe ob jectofm uchof GraphTheory?For visualizationpurp oses?it isw ellestablishedthat
edge crossingssigni?can tlyinhibitreadability ????? andmany algorithmsforconstructingplanar
Ahierarchicalgraph is drawnwithv erticesofa layer onthesame horizontal line?andedges as
curves monotonic inydirection?Ahierarc hicalgraph is hierarchic al planar?h?planar? ifitadmitsa
drawingwithoutedge crossings?Algorithmsfortesting hierarc hicalplanarity are presented in??? ??
F oraclustered graph? theclustering structureis representedbya closedcurvethatde?nesa
region?The regioncon tainsthedrawing ofallthev erticeswhichb elongtothat cluster?Aclustered
graph iscomp oundplanar?c?planar? if it admitsa dra wingwithnoedge crossings oredge?region
crossings? Algorithms for testingcomp oundplanarity arepresented in??? ? ????
The hierarchicalstructure inhierarchical graphs imposes constrain tsonthey ?coordinate?since
allv erticesof thesamelay er has tobe drawnon exactlythesame horizontalline?Ho wever? there
areno constrain tsonthe otherdimension?that is?the x?co ordinate? Theclustering structurein
clustered graphscanbeview edasconstrain tsonboth dimensions?that is?all v erticesofthesame
cluster are restrictedtoaregion?
One of thebasicgraph drawingconv entions consists ofrepresen tingedgesas straight?lineseg?
ments?Thestraight?linedrawingconven tionis widely usedin visualization?Graphdra wingsystems
suchas theGraphLayoutToolkit ?????GraphEd ????andDiagramServer???containamo dule
forcreating straight?line drawingsofclassical graphs?Intuitiv ely? the eyecanfollowa straigh t?line
easily?Sugiyama????andBatini ????liststraightnessof linesasan importantaim for graphdra wing
algorithms?This in tuition hasb een con?rmedbyhumanexperiments???? ????More importantly ?
thereareseveral general methodsfor drawinggraphswhichb eginby addingdummyv erticeson
edges and thenapplya straigh t?linedrawing algorithm? This demandhas spawneda considerable
amount of attention to straight?line dra wings inthe researchcommunit y?
For classicalgraphs? itis well knownthateveryplanargraphadmitsastraight?line drawing
mitsaplanarstraight?linedrawingwhereallthefaceboundaries aredrawnascon vexpolygons?
ings??The?rstquestionweaddressisoffundamental signi?cance fordrawing hierarchicalgraphs?
do esev eryplanarhierarchicalgraph admitaplanar straight?linehierarc hicaldra wing?While many
algorithms havebeendev elop ed todra whierarchicalgraphs??? ?????? ??? ??? ??? ????they allintro?
duceb ends to route theedges?and the basicproblem of planar straight?line dra wingshas notb een
solv edcompletely? Ithasbeensho wnby diBattistaandTamassia ???that every planarst?graph
admits anupward dr awing?thatis?adra wingwhereallarcsaredrawnasstraight?linesegmen ts
pointing upward?How ever?theproblem for hierarchicalgraphs isdi?eren t?because we have more
constraints?verticesof the samelay ershouldbe drawn onthe samehorizontalline andthe layers
shouldbe an equaldistanceapart?Amethod toconstruct straight?linedra wingsof planarhier?
arc hicalgraphsw aspresen tedbyEades? Lin andT amassia??? ??They useatec hniquesimilarto
that in????byTutte??ndingthe positionfor everyvertex in aglobalmanner?Intheiralgorithm?
dummyverticesare added totransform anedge thatspans morethantwola yers toasequence of
edges?each of whichspanstwo consecutivelayers?In fact?by the conv exityof thedra wing?the
dummyv erticesdo notpro ducebends?Theproblemwith thealgorithmof????isthatitonlyworks
The second question addressedin thispaper isforclusteredgraphs? do es everyplanarclustered
graphadmitaplanar straight?line drawingwithclusters drawn asconvexp olygons?Analgorithm
forstraigh t?linedra wingof clusteredgraphs has b eenpresented byFeng? CohenandEades??? ??
Again?ho wev er?itonlyappliesto asp ecialclass ofgraphs?Inparticular?itonly appliestographs
with acertain strong connectivity property?
straight?linedrawing?A lineartime algorithmthat pro duces suchdrawingsis presen ted?We
introducetheclustered graph modelinsection??Insection??weshowthateveryplanarclustered
Inthis sectionweintroducetheterminology?andsomefundamen talpropertiesofhieorarc hical
Adirectededgewithatailuandaheadvisdenotedby?u?v??Ahier archical graphH?
?V?A??? k?consistsofadirectedgraph?V?A??apositive integerk? and?for eachv ertexu? an
integer??u??????????k? withtheprop ertythat if?u? v??A?then ??u????v?? F or??i?k?
the setfuj??u??igistheithlayer ofHand isdenotedbyL
?Thesp anofan edge?u? v?is
??v????u?? Anedgeofspan greaterthanone islong? andahierarchicalgraph withnolongedges
AdrawingofagraphG??V?E?assignsaposition p?v???x?v??y?v ?? toeachvertex v?V
anda curvejoiningp?u?andp?v?toeach edge?u?v??E?Ahierarc hicalgraph isconventionally
on thehorizon tal liney?i?that is?y?v????v?for all verticesv??and edges
as curv esmonotoniciny direction?If no pairofnonincidentedges intersect inthe drawing?thenwe
say itisahier archic alplanar ?h?planar? drawing?Note thata non?proper hierarchicalgraph canbe
transformedin toaprop er hierarchical graphbyadding dummyvertices onlong edges?It iseasily
sho wn thatanon?proper hierarchicalgraph is h?planarifand onlyifthecorrespondingproper
de?nedbytheorderingofverticesoneachlay erof thegraph? Notethateverysuchembedding
hierarchicaldrawing? thatis?a drawing where edgesare dra wnas straigh t?linesegmen ts? How ever?
fornon?properhierarchical graphs?the problem isnot trivial?since nob endsareallowed onlong
Aplane gr aphrefers toaplanargraphem bedded inthe plane?In otherw ords?aplane graph
con tains a planargraphand aplanarembeddingwithaspeci?ed external face?Wecallaplane
emb eddedhierarc hicalgraphahierarchic alplane graph?Ifahierarc hicalplane graphhasonly one
planegraphisaconnectedgraph?anditssources and sinktm ustlieonthebottomlayerand
Theemb eddingofahierarchical?stplanegraphHdetermines?foreveryv ertexv?aleft?right
?v?andthe leftdown?neighb or
?v? ofv arede?nedanalogously ?
Hierarc hicalgraphs aredirected graphs and thuswecanborrowmuchofthe standardtermi?
nology ofgraphtheory ???? The terms?path???cycle??and ?biconnectivity??whenapplied toa
directedgraph inthispaper? referto theunderlyingundirectedgraph?Todenoteacycle ofa
planegraph?we usethe sequenceofverticesonthecycle inclockwise order?Fora cycleor path
??an edgeb etweentwo non?consecutivev erticesinPiscalled achord ofP?see
Figure??a??? Acycle orpathis called chordless ifit hasno chord?In hierarc hicalgraphs?edges
aredirectedfrom a lowerla y ertoahigher layer?Apath iscalledmonotonic ifthedirectionsof the
edgesdonot c hangealongthepath?In otherw ords?a pathis monotonic if thelayerincreases?or
Thefollowinglemma givessomebasicprop erties of thebiconnectedcomponen ts of hierarchical?
stplane graphs?Essen tially it impliesthat? forhierarc hical?stplane graphs?we canrestrict our
attention tothe biconnectedcomp onents?
Lemma?LetH??V?A? ??k?beahierar chical?stplanegr aph?then?