LooPindex - 12^1

$12^{1}_{149}$ - Minimal pinning sets

(y-axis: cardinality)

The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.82049
- on average over minimal pinning sets: 2.17361
- on average over optimal pinning sets: 2.125

Pin label	Pin color	Regions	Cardinality	Degree sequence	Mean-degree
A (optimal)	•	{1, 2, 4, 5, 6, 7, 8, 9}	8	[2, 2, 2, 2, 2, 2, 2, 3]	2.12
a (minimal)	•	{1, 3, 4, 5, 6, 7, 8, 9, 11}	9	[2, 2, 2, 2, 2, 2, 2, 3, 3]	2.22

Cardinality	Optimal pinning sets	Minimal suboptimal pinning sets	Nonminimal pinning sets	Averaged mean-degree
8	1	0	0	2.12
9	0	1	4	2.47
10	0	0	8	2.85
11	0	0	5	3.16
12	0	0	1	3.33
Total	1	1	18

Plantri embedding: [[1,1,2,2],[0,3,3,0],[0,4,4,0],[1,5,5,1],[2,6,7,2],[3,8,8,3],[4,8,9,7],[4,6,9,9],[5,9,6,5],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[19,8,20,9],[10,2,11,1],[7,18,8,19],[2,12,3,11],[17,6,18,7],[12,16,13,15],[3,15,4,14],[5,16,6,17],[13,5,14,4]]
Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (1,10,-2,-11)(14,3,-15,-4)(16,5,-17,-6)(8,19,-9,-20)(11,20,-12,-1)(12,9,-13,-10)(2,13,-3,-14)(4,15,-5,-16)(6,17,-7,-18)(18,7,-19,-8)
- Edge permutation $\epsilon=$ (-1,1)(-2,2)(-3,3)(-4,4)(-5,5)(-6,6)(-7,7)(-8,8)(-9,9)(-10,10)(-11,11)(-12,12)(-13,13)(-14,14)(-15,15)(-16,16)(-17,17)(-18,18)(-19,19)(-20,20)
- Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-11)(-2,-14,-4,-16,-6,-18,-8,-20,11)(-3,14)(-5,16)(-7,18)(-9,12,20)(-10,1,-12)(-13,2,10)(-15,4)(-17,6)(-19,8)(3,13,9,19,7,17,5,15)