Graph Drawing Introduction 2005/2006. Graph Drawing: Introduction2 Contents Applications of graph drawing Planar graphs: some theory Different types of

Graph Drawing

Introduction

2005/2006

Graph Drawing: Introduction2

Contents

• Applications of graph drawing

• Planar graphs: some theory

• Different types of drawings


Applications

• Earliest graph drawings: family trees • VLSI / circuit design

– First major application of automated graph drawing

– Floor planning

• Aesthetically pleasing graph drawings– Visualizing molecules, organisation diagrams, UML-

schemes, database designs (ER-diagrams), questionnaires,


Planar graphs

• Can be drawn on the plane without crossings

• Most road networks

• From models of geometric problems– E.g. Delauney triangulations of point sets


Plane graph

• Plane graph = planar graph given with fixed embedding in the plane

• A planar graph has several embeddings– Some are topologically equivalent, some not.


Some notions

• Faces• Exterior face• Interior faces• Each edge is incident to 2 faces• Boundary of a face• Outer boundary• Outer vertex, outer edge• Inner vertex, inner edge


Equivalent embeddings

• For each face in one embedding, there is a face in the other embedding with the same boundary.

• 3-connected planar graphs have a unique embedding– A graph is 3-connected, if and only if removing any 2

vertices still gives a connected graph

• We can get equivalent embeddings with any face chosen as exterior face


The smallest graphs that are not planar

• K5, K3,3


Kuratowski / Wagner

• A graph is planar if and only if it does not contain the K5 and the K3,3 as a homeomorphic subgraph / as a minor.

• Minor: H is a minor of G, if H can be obtained from G by a series of 0 or more deletions of vertices, deletions of edges, and contraction of edges.

• Does not yield fast recognition algorithm!


Euler’s theorem

• Let G be a connected plane graph with n vertices, m edges, and f faces. Then n + f – m = 2.

• Proof. By induction. – True if m=0.– If G has a circuit, then delete an edge that is part of a

circuit: this decreases f by one and m by one. n stays the same. IH.

– If G has a vertex v of degree 1, then delete v: this decreases n by one, m by one, while f stays the same. IH.


Euler’s theoremCorollaries

• If G is a connected plane graph with no parallel edges and no self-loops, with n > 1, then m 3n-6.– Every face `has’ at least three edges; each edge

`is on’ two faces, or twice on the same face.

• Every plane graph with no parallel edges and no self-loops has a vertex of degree at most 5.


Maximal planar graph

• G=(V,E) (without parallel edges and self-loops) is a maximal planar graph, if – G is planar– For each two vertices v, w, v w, {v,w} E:

• G + {v,w} is not a planar graph

• Often also called: triangulated– Each face is a triangle

• |E| = 3|V| - 6, for maximal planar graphs


Duality

• The dual G* of a plane graph G– A vertex in G* for each face of G– An edge in G* when faces share an edge in G

(G*)* = G(G*)* = G


Testing planarity

• Detailed algorithms can test in O(n) time if a graph G=(V,E) is planar– And find for each vertex a clockwise ordering

of the outgoing edges as they can appear in an embedding in the plane


Types of drawings

• Vertices can be represented as– Points– Circles or rectangles

• Edges can be represented as– Straight lines– Curves– Lines with bends– Sequences of horizontal and vertical line segments– Implicitly, by adjacency of rectangles representing

vertices


Planar drawing

• Vertices are points in de plane

• Edges are curves between endpoints

• No edges cross– Many different types…


Polyline drawings

• Each edge is a polygonal chain

• Each edge is a chain of horizontal and vertical line segments (orthogonal drawing)


Straight line drawing

• Each edge is a straight line.

• Theorem: a planar graph has a plane embedding where each edge is a straight line. (Wagner (1936); Fáry (1948); Stein (1951))

Not astraight

linedrawing

A straightline

drawing


Box orthogonal drawings

• A graph with a vertex of degree more than 4 does not have an orthogonal drawing

• In a box orthogonal drawing a vertex is represented by a rectangle (box)


Rectangular drawing

• Vertices are points• Edges are a vertical or

horizontal line• No crossings• Each face is a

rectangle• Generalization: box

rectangular drawing


Grid drawing

• Each vertex and each bend is on a grid point– Prevents drawings to

have many points in a very small area

– When drawing on raster device (e.g., screen)

– Area minimization


Visibility drawing

• Each vertex is horizontal line segment

• Each edge is vertical line segment between segments of its vertices


Properties of drawings

• Area (e.g. for grid drawings)

• Number of bends– In total; maximum per edge

• Number of crossings (if graph is not planar)

• Aspect ratio (length of longest edge / length of shortest edge)

• Shape of faces (e.g., rectangles, convexity)

• Symmetry (drawing isomorphic parts in the same or mirrored ways)

• Angular resolution (angles between adjacent edges)

• Beauty?


A data structure for a plane graph

• Adjacency lists give edges in clockwise order

• For easily moving around in the graph:– Adjacency lists are doubly linked– The two entries for an edge are linked to each

other• Enables listing all vertices or edges on a fast in a

fast way


Next

• Read chapters 1, 2 and 3 from Planar Graph Drawing

• Schedule for this year

Documents

Graph Drawing Introduction 2005/2006. Graph Drawing: Introduction2 Contents Applications of graph drawing Planar graphs: some theory Different types of