# Graph Drawing - Trees

Graph Drawing
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Graph Drawing Tutorial
Isabel F. Cruz
Worcester Polytechnic Institute
Roberto Tamassia
Brown University
Graph Drawing
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Introduction
Graph Drawing
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Graph Drawing

models, algorithms, and systems for the
visualization of graphs and networks

applications to software engineering (class
hierarchies), database systems (ER-
diagrams), project management (PERT
diagrams), knowledge representation (isa

hierarchies), telecommunications (ring
covers), WWW (browsing history) ...
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Graph Drawing
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orthogonal drawing
bend
Drawing Conventions

general constraints on the geometric
representation of vertices and edges
polyline drawing
planar straight-line drawing
Graph Drawing
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strong visibility representation
planar othogonal straight-line drawing
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abc
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d
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f
a
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Drawing Conventions
Graph Drawing
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Drawing Conventions

directed acyclic graphs are usually drawn
in such a way that all edges “ﬂow” in the
same direction, e.g., from left to right, or
from bottom to top

such upward drawings effectively
visualize hierarchical relationships, such
as covering digraphs of ordered sets

not every planar acyclic digraph admits a
planar upward drawing
Graph Drawing
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Resolution

display devices and the human eye have
ﬁnite resolution

examples of resolution rules:

integer coordinates for vertices and
bends (grid drawings)

prescribed minimum distance between
vertices

prescribed minimum distance between
vertices and nonincident edges

prescribed minimum angle formed by
consecutive incident edges (angular
resolution)
Graph Drawing
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Angular Resolution
• The angular resolution
ρ
of a straight-
line drawing is the smallest angle
formed by two edges incident on the
same vertex
• High angular resolution is desirable
in visualization applications and in the
design of optical communication
networks.
•Atrivial upper bound on the angular
resolution is
where d is the maximum vertex degree.
ρ ≤

d
------
Graph Drawing
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Aesthetic Criteria

some drawings are better than others in
conveying information on the graph

aesthetic criteria attempt to
general optimization goals
Examples

minimize crossings

minimize area

minimize bends (in orthogonal drawings)

minimize slopes (in polyline drawings)

maximize smallest angle

maximize display of symmetries
Graph Drawing
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in general, one cannot simultaneously
optimize two aesthetic criteria
Complexity Issues

testing planarity takes linear time

testing upward planarity is NP-hard

minimizing crossings is NP-hard

minimizing bends in planar orthogonal
drawing:

NP-hard in general

polynomial time for a ﬁxed embedding
min # crossings max symmetries
Graph Drawing
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Beyond Aesthetic Criteria
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Constraints

knowledge about the semantics of the
speciﬁc graph (e.g., place “most
important” vertex in the middle)

input to a graph drawing algorithm
Examples

place a given vertex in the “middle” of
the drawing

place a given vertex on the external
boundary of the drawing

draw a subgraph with a prescribed
“shape”

keep a group of vertices “close” together

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