1. Graph Drawing Introduction 2005/2006

2. Graph Drawing: Introduction2 Contents Applications of graph drawing Planar graphs: some theory Different types of drawings

3. Graph Drawing: Introduction3 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,

4. Graph Drawing: Introduction4 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

5. Graph Drawing: Introduction5 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.

6. Graph Drawing: Introduction6 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

7. Graph Drawing: Introduction7 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

8. Graph Drawing: Introduction8 The smallest graphs that are not planar K 5, K 3,3

9. Graph Drawing: Introduction9 Kuratowski / Wagner A graph is planar if and only if it does not contain the K 5 and the K 3,3 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!

10. Graph Drawing: Introduction10 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.

11. Graph Drawing: Introduction11 Euler’s theorem Corollaries 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.

12. Graph Drawing: Introduction12 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

13. Graph Drawing: Introduction13 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

14. Graph Drawing: Introduction14 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

15. Graph Drawing: Introduction15 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

16. Graph Drawing: Introduction16 Planar drawing Vertices are points in de plane Edges are curves between endpoints No edges cross –Many different types…

17. Graph Drawing: Introduction17 Polyline drawings Each edge is a polygonal chain Each edge is a chain of horizontal and vertical line segments (orthogonal drawing)

18. Graph Drawing: Introduction18 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); Fá ry (1948); Stein (1951)) Not a straight line drawing A straight line drawing

19. Graph Drawing: Introduction19 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)

20. Graph Drawing: Introduction20 Rectangular drawing Vertices are points Edges are a vertical or horizontal line No crossings Each face is a rectangle Generalization: box rectangular drawing

21. Graph Drawing: Introduction21 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

22. Graph Drawing: Introduction22 Visibility drawing Each vertex is horizontal line segment Each edge is vertical line segment between segments of its vertices

23. Graph Drawing: Introduction23 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?

24. Graph Drawing: Introduction24 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

25. Graph Drawing: Introduction25 Next Read chapters 1, 2 and 3 from Planar Graph Drawing Schedule for this year