(A) combinatorics of Proposition 8.5 for 5 <= m <= 10 and all admissible I
  [ok] m=5: 10 admissible sets I, 220 connected dominating sets containing q: each contains i-1 or i+1 for every i in I; cliques through q are {q}, {q,j}, {q,j,j+1}
  [ok] m=6: 15 admissible sets I, 636 connected dominating sets containing q: each contains i-1 or i+1 for every i in I; cliques through q are {q}, {q,j}, {q,j,j+1}
  [ok] m=7: 28 admissible sets I, 2184 connected dominating sets containing q: each contains i-1 or i+1 for every i in I; cliques through q are {q}, {q,j}, {q,j,j+1}
  [ok] m=8: 44 admissible sets I, 6528 connected dominating sets containing q: each contains i-1 or i+1 for every i in I; cliques through q are {q}, {q,j}, {q,j,j+1}
  [ok] m=9: 75 admissible sets I, 20664 connected dominating sets containing q: each contains i-1 or i+1 for every i in I; cliques through q are {q}, {q,j}, {q,j,j+1}
  [ok] m=10: 120 admissible sets I, 62320 connected dominating sets containing q: each contains i-1 or i+1 for every i in I; cliques through q are {q}, {q,j}, {q,j,j+1}

(B) m = 5 in full: the minimal sets not contained in a join of Gamma_I (q = vertex 0)
  I = (1,): Lambda_I has 9 edges; 35 connected dominating sets, minimal ones: [[0, 2], [0, 5], [1, 2, 3], [1, 2, 5], [1, 4, 5], [2, 3, 4], [3, 4, 5]]
     Gamma_I: q adjacent to [1]; edges [(0, 1), (1, 3), (1, 4), (2, 4), (2, 5), (3, 5)]
  I = (1, 3): Lambda_I has 8 edges; 31 connected dominating sets, minimal ones: [[0, 2], [0, 4, 5], [1, 2, 3], [1, 2, 5], [1, 4, 5], [2, 3, 4], [3, 4, 5]]
     Gamma_I: q adjacent to [1, 3]; edges [(0, 1), (0, 3), (1, 3), (1, 4), (2, 4), (2, 5), (3, 5)]

(C) planar picture and covers
  [ok] m=5: D_j is a disc with D_j n P = {s_(j-1), s_j}
  [ok] m=5: consecutive discs: 2 crossings, 4 components each with a puncture, int D_i - bd D_(i+-1) = two discs with one puncture each (hypothesis of Lemma 8.4); non-consecutive discs disjoint
  [ok] m=5: full arrangement: 10 crossings, 12 faces, discs with <= 1 puncture
  [ok] m=5: all complementary components are discs with <= 1 puncture exactly for the 11 sets Y containing some Y_j = V - {j,j+1} (of 31 non-empty sets)
  [ok] m=5 d= 5 w=[1, 1, 1, 1, 1]: genus 6; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 10 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [2, 4] (>= 2); (P) for the cliques through q
  [ok] m=5 d= 7 w=[1, 1, 1, 1, 3]: genus 9; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 10 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [3, 6] (>= 2); (P) for the cliques through q
  [ok] m=5 d= 9 w=[1, 1, 1, 1, 5]: genus 12; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 10 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [3, 4, 7, 8] (>= 2); (P) for the cliques through q
  [ok] m=5 d=15 w=[1, 1, 1, 1, 11]: genus 21; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 10 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [6, 7, 13, 14] (>= 2); (P) for the cliques through q
  [ok] m=6: D_j is a disc with D_j n P = {s_(j-1), s_j}
  [ok] m=6: consecutive discs: 2 crossings, 4 components each with a puncture, int D_i - bd D_(i+-1) = two discs with one puncture each (hypothesis of Lemma 8.4); non-consecutive discs disjoint
  [ok] m=6: full arrangement: 12 crossings, 14 faces, discs with <= 1 puncture
  [ok] m=6: all complementary components are discs with <= 1 puncture exactly for the 13 sets Y containing some Y_j = V - {j,j+1} (of 63 non-empty sets)
  [ok] m=6 d= 5 w=[1, 1, 1, 1, 3, 3]: genus 8; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 15 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [4, 6] (>= 2); (P) for the cliques through q
  [ok] m=6 d= 7 w=[1, 1, 1, 1, 1, 2]: genus 12; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 15 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [6, 9] (>= 2); (P) for the cliques through q
  [ok] m=6 d= 9 w=[1, 1, 1, 1, 1, 4]: genus 16; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 15 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [8, 12] (>= 2); (P) for the cliques through q
  [ok] m=6 d=15 w=[1, 1, 1, 1, 4, 7]: genus 28; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 15 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [12, 14, 19, 21] (>= 2); (P) for the cliques through q
     outside the hypothesis: m=6, I=(1, 3, 5) (2|I| = m), d=5, w=[1, 1, 1, 1, 3, 3]: U_q has 1 component(s), chi -5, 3 boundary curves, genus 2
     outside the hypothesis: m=6, I=(1, 3, 5) (2|I| = m), d=9, w=[1, 1, 1, 1, 1, 4]: U_q has 1 component(s), chi -9, 3 boundary curves, genus 4
     outside the hypothesis: m=6, I=(1, 3, 5) (2|I| = m), d=15, w=[1, 1, 1, 1, 4, 7]: U_q has 1 component(s), chi -15, 7 boundary curves, genus 5
     m=6, I=(1,3,5), d=9, w=[1, 2, 1, 2, 1, 2] (consecutive sums 3): U_q has 3 components
  [ok] m=7: D_j is a disc with D_j n P = {s_(j-1), s_j}
  [ok] m=7: consecutive discs: 2 crossings, 4 components each with a puncture, int D_i - bd D_(i+-1) = two discs with one puncture each (hypothesis of Lemma 8.4); non-consecutive discs disjoint
  [ok] m=7: full arrangement: 14 crossings, 16 faces, discs with <= 1 puncture
  [ok] m=7: all complementary components are discs with <= 1 puncture exactly for the 15 sets Y containing some Y_j = V - {j,j+1} (of 127 non-empty sets)
  [ok] m=7 d= 5 w=[1, 1, 1, 1, 2, 1, 3]: genus 10; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 28 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [4, 6, 8] (>= 2); (P) for the cliques through q
  [ok] m=7 d= 7 w=[1, 1, 1, 1, 1, 1, 1]: genus 15; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 28 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [6, 9, 12] (>= 2); (P) for the cliques through q
  [ok] m=7 d= 9 w=[1, 1, 1, 1, 1, 2, 2]: genus 20; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 28 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [6, 7, 8, 10, 11, 12, 15, 16] (>= 2); (P) for the cliques through q
  [ok] m=7 d=15 w=[1, 1, 1, 1, 1, 2, 8]: genus 35; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 28 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [12, 13, 19, 20, 21, 26, 27, 28] (>= 2); (P) for the cliques through q
  [ok] m=8: D_j is a disc with D_j n P = {s_(j-1), s_j}
  [ok] m=8: consecutive discs: 2 crossings, 4 components each with a puncture, int D_i - bd D_(i+-1) = two discs with one puncture each (hypothesis of Lemma 8.4); non-consecutive discs disjoint
  [ok] m=8: full arrangement: 16 crossings, 18 faces, discs with <= 1 puncture
  [ok] m=8: all complementary components are discs with <= 1 puncture exactly for the 17 sets Y containing some Y_j = V - {j,j+1} (of 255 non-empty sets)
  [ok] m=8 d= 5 w=[1, 1, 1, 1, 1, 1, 1, 3]: genus 12; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 44 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [6, 8, 10] (>= 2); (P) for the cliques through q
  [ok] m=8 d= 7 w=[1, 1, 1, 1, 1, 1, 3, 5]: genus 18; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 44 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [9, 12, 15] (>= 2); (P) for the cliques through q
  [ok] m=8 d= 9 w=[1, 1, 1, 1, 1, 1, 1, 2]: genus 24; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 44 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [11, 12, 15, 16, 19, 20] (>= 2); (P) for the cliques through q
  [ok] m=8 d=15 w=[1, 1, 1, 1, 1, 1, 1, 8]: genus 42; X_j genus>=2 with gcd(d,w_(j-1)+w_j) boundary curves; for all 44 admissible I: U_q connected, chi as in the text, boundary = sum b_i, genus in [20, 21, 27, 28, 34, 35] (>= 2); (P) for the cliques through q
     outside the hypothesis: m=8, I=(1, 3, 5, 7) (2|I| = m), d=5, w=[1, 1, 1, 1, 1, 1, 1, 3]: U_q has 1 component(s), chi -10, 4 boundary curves, genus 4
     outside the hypothesis: m=8, I=(1, 3, 5, 7) (2|I| = m), d=9, w=[1, 1, 1, 1, 1, 1, 1, 2]: U_q has 1 component(s), chi -18, 6 boundary curves, genus 7
     outside the hypothesis: m=8, I=(1, 3, 5, 7) (2|I| = m), d=15, w=[1, 1, 1, 1, 1, 1, 1, 8]: U_q has 1 component(s), chi -30, 6 boundary curves, genus 13

RESULT: ALL CHECKS PASSED
