Drawing Planar Graphs
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1 Mathematics for Computer Science MIT 6.042J/18.062J Drawing Planar Graphs Copyright Albert R. Meyer, October 8,
2 Planar Graphs A graph is planar if there is a way to draw it in the plane without edges crossing. Copyright Albert R. Meyer, October 8,
3 Planar Graphs Maps are 2-connected planar graphs Copyright Albert R. Meyer, October 8,
4 Planar Graphs Maps are 2-connected planar graphs General connected planar graphs may have Copyright Albert R. Meyer, October 8,
5 Planar Graphs Maps are 2-connected planar graphs General connected planar graphs may have dongles Copyright Albert R. Meyer, October 8,
6 Planar Graphs Maps are 2-connected planar graphs General connected planar graphs may have dongles cross bars Copyright Albert R. Meyer, October 8,
7 A Planar Graph Copyright Albert R. Meyer, October 8,
8 A Planar Graph with 3 faces Copyright Albert R. Meyer, October 8,
9 A Planar Graph with 3 faces Copyright Albert R. Meyer, October 8,
10 A Planar Graph with 3 faces (wait! also the outer face) Copyright Albert R. Meyer, October 8,
11 A Planar Graph with 43 faces ( wait! also the outer face) Copyright Albert R. Meyer, October 8,
12 Drawing a Planar Graph draw it edge by edge, starting with a single vertex Copyright Albert R. Meyer, October 8,
13 Drawing a Planar Graph draw it edge by edge, starting with a single vertex graph Copyright Albert R. Meyer, October 8,
14 Drawing a Planar Graph and record faces while drawing graph Copyright Albert R. Meyer, October 8,
15 Drawing a Planar Graph and record faces while drawing graph faces Copyright Albert R. Meyer, October 8,
16 Planar Graphs and record faces while drawing graph faces Copyright Albert R. Meyer, October 8,
17 Planar Graphs and record faces while drawing graph faces Copyright Albert R. Meyer, October 8,
18 Planar Graphs and record faces while drawing graph faces Copyright Albert R. Meyer, October 8,
19 Planar Graphs and record faces while drawing graph faces Copyright Albert R. Meyer, October 8,
20 Planar Graphs and record faces while drawing graph faces (the outer face) Copyright Albert R. Meyer, October 8,
21 Planar Graphs and record faces while drawing graph faces (the outer face) Copyright Albert R. Meyer, October 8,
22 Planar Graphs and record faces while drawing graph faces (the outer face) Copyright Albert R. Meyer, October 8,
23 Planar Graphs and record faces while drawing graph faces (the outer face) Copyright Albert R. Meyer, October 8,
24 Planar Graphs and record faces while drawing graph faces (the outer face) Copyright Albert R. Meyer, October 8,
25 Planar Graphs and record faces while drawing graph faces (the outer face) Copyright Albert R. Meyer, October 8,
26 Recursive Definition of Faces Precise rules defining the cycles that are the face boundaries of a Planar Drawing: Copyright Albert R. Meyer, October 8,
27 Recursive Definition of Faces Start with a vertex Copyright Albert R. Meyer, October 8,
28 Recursive Definition of Faces Start with a vertex There is one face, whose boundary is the 0-length cycle consisting of this vertex. Copyright Albert R. Meyer, October 8,
29 Recursively adding an edge to a drawing Two cases for connected graph: 1) Attach edge from vertex on a face to a new vertex. 2) Attach edge between nonadjacent vertices on a face. Copyright Albert R. Meyer, October 8,
30 Face Creation Rule 1 1) choose vertex, v, on a face boundary Copyright Albert R. Meyer, October 8,
31 Face Creation Rule 1 1) choose vertex, v, on a face boundary v Copyright Albert R. Meyer, October 8,
32 Face Creation Rule 1 1) choose vertex, v, on a face boundary path x v Copyright Albert R. Meyer, October 8,
33 Face Creation Rule 1 1) choose vertex, v, on a face boundary path x v face boundary vxv Copyright Albert R. Meyer, October 8,
34 Face Creation Rule 1 1) choose vertex, v, on a face boundary v w Create new vertex, w, Copyright Albert R. Meyer, October 8,
35 Face Creation Rule 1 1) choose vertex, v, on a face boundary w v Create new vertex, w, and add edge v--w Copyright Albert R. Meyer, October 8,
36 Face Creation Rule 1 path x w v old face boundary vxv Copyright Albert R. Meyer, October 8,
37 Face Creation Rule 1 path x w v new face boundary Copyright Albert R. Meyer, October 8,
38 Face Creation Rule 1 path x w v new face boundary Copyright Albert R. Meyer, October 8,
39 Face Creation Rule 1 path x w v new face boundary vwvxv Copyright Albert R. Meyer, October 8,
40 Face Creation Rules nothing else changes new face boundary vwvxv Copyright Albert R. Meyer, October 8,
41 Recursive Face Creation Rule 2 2) choose vertices v, w on a face boundary Copyright Albert R. Meyer, October 8,
42 Face Creation Rule 2 2) choose vertices v, w on a face boundary w v Copyright Albert R. Meyer, October 8,
43 Face Creation Rule 2 2) choose vertices v, w on a face boundary w v with v, w, not adjacent Copyright Albert R. Meyer, October 8,
44 Face Creation Rule 2 2) choose vertices v, w on a face boundary w y v x face boundary vywxv Copyright Albert R. Meyer, October 8,
45 Face Creation Rule 2 2) choose vertices v, w on a face boundary w v and add edge v--w Copyright Albert R. Meyer, October 8,
46 Face Creation Rule 2 w x y v old face boundary vywxv Copyright Albert R. Meyer, October 8,
47 Face Creation Rule 2 w x y v old face boundary vywxv Copyright Albert R. Meyer, October 8,
48 Face Creation Rule 2 w v splits into 2 faces: Copyright Albert R. Meyer, October 8,
49 Face Creation Rule 2 w w v v splits into 2 faces: Copyright Albert R. Meyer, October 8,
50 Face Creation Rule 2 w w x v v splits into 2 faces: vwxv Copyright Albert R. Meyer, October 8,
51 Face Creation Rule 2 w w y x v v splits into 2 faces: vwxv, vywv Copyright Albert R. Meyer, October 8,
52 Face Creation Rules nothing else changes splits into 2 faces: vwxv, vywv Copyright Albert R. Meyer, October 8,
53 Recursive Definition of Faces Every connected planar drawing is obtained by starting with a single vertex, and repeatedly applying Rules 1 & 2. Copyright Albert R. Meyer, October 8,
54 Induction on Drawings Properties of planar drawings like Euler s formula can be proved by induction on the number of rule applications used to create a drawing. Copyright Albert R. Meyer, October 8,
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