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
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models, algorithms, and systems for the
visualization of graphs and networks
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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
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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
gf
abc
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f
a
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Drawing Conventions
Graph Drawing
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Drawing Conventions
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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
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such upward drawings effectively
visualize hierarchical relationships, such
as covering digraphs of ordered sets
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not every planar acyclic digraph admits a

planar upward drawing
Graph Drawing
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Resolution
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display devices and the human eye have
finite resolution
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examples of resolution rules:
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integer coordinates for vertices and
bends (grid drawings)
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prescribed minimum distance between
vertices
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prescribed minimum distance between
vertices and nonincident edges
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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 ≤
2π
d
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Graph Drawing
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Aesthetic Criteria
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some drawings are better than others in
conveying information on the graph
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aesthetic criteria attempt to
characterize readability by means of
general optimization goals
Examples
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minimize crossings
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minimize area
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minimize bends (in orthogonal drawings)
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minimize slopes (in polyline drawings)
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maximize smallest angle
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maximize display of symmetries
Graph Drawing
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Trade-Offs
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in general, one cannot simultaneously
optimize two aesthetic criteria
Complexity Issues
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testing planarity takes linear time
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testing upward planarity is NP-hard
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minimizing crossings is NP-hard
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minimizing bends in planar orthogonal
drawing:
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NP-hard in general
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polynomial time for a fixed embedding
min # crossings max symmetries
Graph Drawing
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Beyond Aesthetic Criteria
Graph Drawing

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Constraints
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some readability aspects require
knowledge about the semantics of the
specific graph (e.g., place “most
important” vertex in the middle)
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constraints are provided as additional
input to a graph drawing algorithm
Examples
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place a given vertex in the “middle” of
the drawing
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place a given vertex on the external
boundary of the drawing
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draw a subgraph with a prescribed
“shape”
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keep a group of vertices “close” together