Example I
1a
2a
3a
4a
5a
6a
7a
R+e
7a
6a
5a
4a3a
2a
1a
R
1b
2b
3b
4b
5b
6b
7b
1b
2b
3b4b
5b
6b
7b
Ankur Moitra () Geometric Graph Theory October 26, 2008 36 / 82
Example I
1a
2a
3a
4a
5a
6a
7a
R+e
7a
6a
5a
4a3a
2a
1a
R
1b
2b
3b
4b
5b
6b
7b
1b
2b
3b4b
5b
6b
7b
Ankur Moitra () Geometric Graph Theory October 26, 2008 37 / 82
Example I
1a
2a
3a
4a
5a
6a
7a
R+e
7a
6a
5a
4a3a
2a
1a
R
1b
2b
3b
4b
5b
6b
7b
1b
2b
3b4b
5b
6b
7b
s
t
Ankur Moitra () Geometric Graph Theory October 26, 2008 38 / 82
Example I
1a
2a
3a
4a
5a
6a
7a
R+e
7a
6a
5a
4a3a
2a
1a
R
1b
2b
3b
4b
5b
6b
7b
1b
2b
3b4b
5b
6b
7b
s
t
Ankur Moitra () Geometric Graph Theory October 26, 2008 39 / 82
Example I
1a
2a
3a
4a
5a
6a
7a
R+e
7a
6a
5a
4a3a
2a
1a
R
1b
2b
3b
4b
5b
6b
7b
1b
2b
3b4b
5b
6b
7b
s
t
Ankur Moitra () Geometric Graph Theory October 26, 2008 40 / 82
Example I
1a
2a
3a
4a
5a
6a
7a
R+e
7a
6a
5a
4a3a
2a
1a
R
1b
2b
3b
4b
5b
6b
7b
1b
2b
3b4b
5b
6b
7b
s
t
Ankur Moitra () Geometric Graph Theory October 26, 2008 41 / 82
Example I
1a
2a
3a
4a
5a
6a
7a
R+e
7a
6a
5a
4a3a
2a
1a
R
1b
2b
3b
4b
5b
6b
7b
1b
2b
3b4b
5b
6b
7b
s
t
Ankur Moitra () Geometric Graph Theory October 26, 2008 42 / 82
Example I
1a
2a
3a
4a
5a
6a
7a
R+e
7a
6a
5a
4a3a
2a
1a
R
1b
2b
3b
4b
5b
6b
7b
1b
2b
3b4b
5b
6b
7b
s
t
Ankur Moitra () Geometric Graph Theory October 26, 2008 43 / 82
Example I
1a
2a
3a
4a
5a
6a
7a
R+e
7a
6a
5a
4a3a
2a
1a
R
1b
2b
3b
4b
5b
6b
7b
1b
2b
3b4b
5b
6b
7b
s
t
Ankur Moitra () Geometric Graph Theory October 26, 2008 44 / 82
Example I
1a
2a
3a
4a
5a
6a
7a
R+e
7a
6a
5a
4a3a
2a
1a
R
1b
2b
3b
4b
5b
6b
7b
1b
2b
3b4b
5b
6b
7b
s
t
Ankur Moitra () Geometric Graph Theory October 26, 2008 45 / 82
Example I
1a
2a
3a
4a
5a
6a
7a
R+e
7a
6a
5a
4a3a
2a
1a
R
1b
2b
3b
4b
5b
6b
7b
1b
2b
3b4b
5b
6b
7b
s
t
Ankur Moitra () Geometric Graph Theory October 26, 2008 46 / 82
Example I
1a
2a
3a
4a
5a
6a
7a
R+e
7a
6a
5a
4a3a
2a
1a
R
1b
2b
3b
4b
5b
6b
7b
1b
2b
3b4b
5b
6b
7b
s
t
Ankur Moitra () Geometric Graph Theory October 26, 2008 47 / 82
Example I
1a
2a
3a
4a
5a
6a
7a
R+e
7a
6a
5a
4a3a
2a
1a
R
1b
2b
3b
4b
5b
6b
7b
1b
2b
3b4b
5b
6b
7b
s
t
Ankur Moitra () Geometric Graph Theory October 26, 2008 48 / 82
Example II
4a
1a
2a
3a
R+eR
1b
2b
2c
3b
2d
3c
3d
4b
4c
4d
1a
2a
2b2c 2d
1b
1c1d
3a
3b3c
3d
4a
4b4c4d
1d
1c
Ankur Moitra () Geometric Graph Theory October 26, 2008 49 / 82
Example II
4d
3c
3d
4c
R+eR
1b
2b
2c
3b
2d
3c
3d
4b
4c
4d
1a
2a
2b
1b
3a
3b
4a
4b
1d
1c1a
2a
3a
4a
1c1d
2c
2d
Ankur Moitra () Geometric Graph Theory October 26, 2008 50 / 82
Example II
4d
3c
3d
4c
R+eR
1b
2b
2c
3b
2d
3c
3d
4b
4c
4d
1a
2a
2b
1b
3a
3b
4a
4b
1d
1c1a
2a
3a
4a
1c1d
2c
2d
Ankur Moitra () Geometric Graph Theory October 26, 2008 51 / 82
Example II
2a
2b 2d
3b
Ankur Moitra () Geometric Graph Theory October 26, 2008 52 / 82
Example II
3b
2a
2b 2d
Ankur Moitra () Geometric Graph Theory October 26, 2008 53 / 82
Example II
3b
2a
2b 2d
Ankur Moitra () Geometric Graph Theory October 26, 2008 54 / 82
Example II
4a
1a
2a
3a
R+eR
1b
2b
2c
3b
2d
3c
3d
4b
4c
4d
1a
2a
2b2c 2d
1b
1c1d
3a
3b3c
3d
4a
4b4c4d
1d
1c
Ankur Moitra () Geometric Graph Theory October 26, 2008 55 / 82
Example II
t
3a
4a
s
R+eR
1b
2b
2c
3b
2d
3c
3d
4b
4c
4d
1a
2a
2b2c 2d
1b
1c1d
3a
3b3c
3d
4a
4b4c4d
1d
1c1a
2a
Ankur Moitra () Geometric Graph Theory October 26, 2008 56 / 82
Example II
t
3a
4a
s
R+eR
1b
2b
2c
3b
2d
3c
3d
4b
4c
4d
1a
2a
2b2c 2d
1b
1c1d
3a
3b3c
3d
4a
4b4c4d
1d
1c1a
2a
Ankur Moitra () Geometric Graph Theory October 26, 2008 57 / 82
Example II
2d
t
s2b
2a
Ankur Moitra () Geometric Graph Theory October 26, 2008 58 / 82
Example II
2d
t
s2b
2a
Ankur Moitra () Geometric Graph Theory October 26, 2008 59 / 82
Example II
t
3a
4a
s
R+eR
1b
2b
2c
3b
2d
3c
3d
4b
4c
4d
1a
2a
2b2c 2d
1b
1c1d
3a
3b3c
3d
4a
4b4c4d
1d
1c1a
2a
Ankur Moitra () Geometric Graph Theory October 26, 2008 60 / 82
Example II
3c
2a
2b 2d
Ankur Moitra () Geometric Graph Theory October 26, 2008 61 / 82
Example II
3c
2a
2b 2d
Ankur Moitra () Geometric Graph Theory October 26, 2008 62 / 82
Example II
t
3a
4a
s
R+eR
1b
2b
2c
3b
2d
3c
3d
4b
4c
4d
1a
2a
2b2c 2d
1b
1c1d
3a
3b3c
3d
4a
4b4c4d
1d
1c1a
2a
Ankur Moitra () Geometric Graph Theory October 26, 2008 63 / 82
Example II
t
3a
4a
s
R+eR
1b
2b
2c
3b
2d
3c
3d
4b
4c
4d
1a
2a
2b2c 2d
1b
1c1d
3a
3b3c
3d
4a
4b4c4d
1d
1c1a
2a
Ankur Moitra () Geometric Graph Theory October 26, 2008 64 / 82
Example II
t
3a
4a
s
R+eR
1b
2b
2c
3b
2d
3c
3d
4b
4c
4d
1a
2a
2b2c 2d
1b
1c1d
3a
3b3c
3d
4a
4b4c4d
1d
1c1a
2a
Ankur Moitra () Geometric Graph Theory October 26, 2008 65 / 82
Example II
t
3a
4a
s
R+eR
1b
2b
2c
3b
2d
3c
3d
4b
4c
4d
1a
2a
2b2c 2d
1b
1c1d
3a
3b3c
3d
4a
4b4c4d
1d
1c1a
2a
Ankur Moitra () Geometric Graph Theory October 26, 2008 66 / 82
Example II
t
3a
4a
s
R+eR
1b
2b
2c
3b
2d
3c
3d
4b
4c
4d
1a
2a
2b2c 2d
1b
1c1d
3a
3b3c
3d
4a
4b4c4d
1d
1c1a
2a
Ankur Moitra () Geometric Graph Theory October 26, 2008 67 / 82
Example II
t
3a
4a
s
R+eR
1b
2b
2c
3b
2d
3c
3d
4b
4c
4d
1a
2a
2b2c 2d
1b
1c1d
3a
3b3c
3d
4a
4b4c4d
1d
1c1a
2a
Ankur Moitra () Geometric Graph Theory October 26, 2008 68 / 82
General Christmas Cactus Graphs
G
Ankur Moitra () Geometric Graph Theory October 26, 2008 69 / 82
General Christmas Cactus Graphs
G
Ankur Moitra () Geometric Graph Theory October 26, 2008 70 / 82
General Christmas Cactus Graphs
G
Ankur Moitra () Geometric Graph Theory October 26, 2008 71 / 82
General Christmas Cactus Graphs
G
Ankur Moitra () Geometric Graph Theory October 26, 2008 72 / 82
General Christmas Cactus Graphs
G
Ankur Moitra () Geometric Graph Theory October 26, 2008 73 / 82
General Christmas Cactus Graphs
G
Ankur Moitra () Geometric Graph Theory October 26, 2008 74 / 82
General Christmas Cactus Graphs
G
Ankur Moitra () Geometric Graph Theory October 26, 2008 75 / 82
General Christmas Cactus Graphs
G
Ankur Moitra () Geometric Graph Theory October 26, 2008 76 / 82
General Christmas Cactus Graphs
G
Ankur Moitra () Geometric Graph Theory October 26, 2008 77 / 82