(1) FILTRATION: 4 5 3 0 35 1 14 34 05 01 13 03 013 04 035 014 034

step  simplex   boundary                 reduction (add earlier reduced columns until low is new)      result
  1   4         {}                                                                                      R=0: positive
  2   5         {}                                                                                      R=0: positive
  3   3         {}                                                                                      R=0: positive
  4   0         {}                                                                                      R=0: positive
  5   35        {5,3}                                                                                   low=3: PAIR (3,35)
  6   1         {}                                                                                      R=0: positive
  7   14        {4,1}                                                                                   low=1: PAIR (1,14)
  8   34        {4,3}                   +R(35)={5,3} -> {4,5}                                           low=5: PAIR (5,34)
  9   05        {5,0}                                                                                   low=0: PAIR (0,05)
 10   01        {0,1}                   +R(14)={4,1} -> {4,0} +R(05)={5,0} -> {4,5} +R(34)={4,5} -> {}  R=0: positive
 11   13        {3,1}                   +R(14)={4,1} -> {4,3} +R(35)={5,3} -> {4,5} +R(34)={4,5} -> {}  R=0: positive
 12   03        {3,0}                   +R(05)={5,0} -> {5,3} +R(35)={5,3} -> {}                        R=0: positive
 13   013       {01,13,03}                                                                              low=03: PAIR (03,013)
 14   04        {4,0}                   +R(05)={5,0} -> {4,5} +R(34)={4,5} -> {}                        R=0: positive
 15   035       {35,05,03}              +R(013)={01,13,03} -> {35,05,01,13}                             low=13: PAIR (13,035)
 16   014       {14,01,04}                                                                              low=04: PAIR (04,014)
 17   034       {34,03,04}              +R(014)={14,01,04} -> {14,34,01,03} +R(013)={01,13,03} -> {14,34,13} +R(035)={35,05,01,13} -> {35,14,34,05,01}  low=01: PAIR (01,034)

P: [('0', '05'), ('01', '034'), ('03', '013'), ('04', '014'), ('1', '14'), ('13', '035'), ('3', '35'), ('5', '34')]
V_P: [('0', '05'), ('03', '013'), ('04', '014'), ('1', '14'), ('3', '35')]
closed V-path in V_P: None
non-incident pair {5,34}: #gradient paths 34 -> 5 in V_P = 1 ['34 3 35 5']
non-incident pair {13,035}: #gradient paths 035 -> 13 in V_P = 1 ['035 03 013 13']
non-incident pair {01,034}: #gradient paths 034 -> 01 in V_P = 2 ['034 04 014 01', '034 03 013 01']

(2) smallest found instance where the earliest non-incident pair has >=2 V'-paths: size 19
order: 3 4 2 0 1 23 13 01 02 12 04 24 14 024 012 124 34 234 134
earliest non-incident pair {02,134}; #V'-paths = 3 ; #V_P-paths = 3
   path 134 34 234 24 024 02
   path 134 14 124 24 024 02
   path 134 14 124 12 012 02
