graphs up to isomorphism, n = 1..7: [1, 2, 4, 11, 34, 156, 1044]

n | non-join | (i) Thm 1.5 without Thm 1.4 | (ii) Thm 1.5 | (iii) Prop 9.3 | chordal || connected non-join | (i) | (ii) | (iii)
1 | 1 | 1 | 1 | 1 | 1 || 0 | 0 | 0 | 0
2 | 1 | 1 | 1 | 1 | 1 || 0 | 0 | 0 | 0
3 | 2 | 2 | 2 | 2 | 2 || 0 | 0 | 0 | 0
4 | 6 | 6 | 6 | 6 | 6 || 1 | 1 | 1 | 1
5 | 21 | 21 | 21 | 21 | 17 || 8 | 8 | 8 | 8
6 | 112 | 93 | 105 | 112 | 67 || 68 | 49 | 61 | 68
7 | 853 | 477 | 578 | 758 | 299 || 662 | 305 | 394 | 567

n=1: covered by Thm 1.5 only through Thm 1.4: 0 ; covered by Prop 9.3 but not by Thm 1.5: 0 ; Thm 1.5 with 4-cycles forbidden as atoms: 1
n=2: covered by Thm 1.5 only through Thm 1.4: 0 ; covered by Prop 9.3 but not by Thm 1.5: 0 ; Thm 1.5 with 4-cycles forbidden as atoms: 1
n=3: covered by Thm 1.5 only through Thm 1.4: 0 ; covered by Prop 9.3 but not by Thm 1.5: 0 ; Thm 1.5 with 4-cycles forbidden as atoms: 2
n=4: covered by Thm 1.5 only through Thm 1.4: 0 ; covered by Prop 9.3 but not by Thm 1.5: 0 ; Thm 1.5 with 4-cycles forbidden as atoms: 6
n=5: covered by Thm 1.5 only through Thm 1.4: 0 ; covered by Prop 9.3 but not by Thm 1.5: 0 ; Thm 1.5 with 4-cycles forbidden as atoms: 21
n=6: covered by Thm 1.5 only through Thm 1.4: 12 ; covered by Prop 9.3 but not by Thm 1.5: 7 ; Thm 1.5 with 4-cycles forbidden as atoms: 95
n=7: covered by Thm 1.5 only through Thm 1.4: 101 ; covered by Prop 9.3 but not by Thm 1.5: 180 ; Thm 1.5 with 4-cycles forbidden as atoms: 495

the 7 graphs on 6 vertices not covered by Theorem 1.5, and how Proposition 9.3 covers them:
  Gamma edges [(0, 3), (0, 4), (1, 4), (1, 5), (2, 5), (3, 5)] | connected True
     complement edges [(0, 1), (0, 2), (0, 5), (1, 2), (1, 3), (2, 3), (2, 4), (3, 4), (4, 5)]
     -> (b+) complement in T0: Lambda(m=5, I=[4])
  Gamma edges [(0, 3), (0, 5), (1, 4), (1, 5), (2, 4), (2, 5)] | connected True
     complement edges [(0, 1), (0, 2), (0, 4), (1, 2), (1, 3), (2, 3), (3, 4), (3, 5), (4, 5)]
     -> (b+) complement in T0: true twin of [clique-sum along a K2 of [K3; C4]]
  Gamma edges [(0, 3), (0, 5), (1, 4), (1, 5), (2, 4), (2, 5), (3, 5)] | connected True
     complement edges [(0, 1), (0, 2), (0, 4), (1, 2), (1, 3), (2, 3), (3, 4), (4, 5)]
     -> (b+) complement in T0: true twin of [clique-sum along a K1 of [C4; K2]] | (b') true twins (in Gamma) added to 1 vertex/vertices of the graph with complement [true twin of [clique-sum along a K1 of [K2; clique-sum along a K1 of [K2; K2]]]]
  Gamma edges [(0, 3), (0, 4), (1, 4), (1, 5), (2, 4), (2, 5), (3, 5)] | connected True
     complement edges [(0, 1), (0, 2), (0, 5), (1, 2), (1, 3), (2, 3), (3, 4), (4, 5)]
     -> (b+) complement in T0: true twin of [C5]
  Gamma edges [(0, 3), (0, 4), (1, 3), (1, 5), (2, 4), (2, 5)] | connected True
     complement edges [(0, 1), (0, 2), (0, 5), (1, 2), (1, 4), (2, 3), (3, 4), (3, 5), (4, 5)]
     -> (b+) complement in T0: prism
  Gamma edges [(0, 3), (0, 4), (1, 3), (1, 5), (2, 4), (2, 5), (3, 5)] | connected True
     complement edges [(0, 1), (0, 2), (0, 5), (1, 2), (1, 4), (2, 3), (3, 4), (4, 5)]
     -> (b+) complement in T0: Lambda(m=5, I=[2, 4])
  Gamma edges [(0, 2), (0, 4), (0, 5), (1, 3), (1, 4), (2, 4), (2, 5), (3, 5)] | connected True
     complement edges [(0, 1), (0, 3), (1, 2), (1, 5), (2, 3), (3, 4), (4, 5)]
     -> (b') true twins (in Gamma) added to 1 vertex/vertices of the graph with complement [C5]

graphs on 7 vertices not covered by Proposition 9.3: 95 (connected: 95)
   by number of edges of Gamma: {7: 10, 8: 18, 9: 27, 10: 21, 11: 10, 12: 7, 13: 2}
   of these, doubles along a star of a connected non-join graph with at most 6 vertices: 4
      Gamma edges [(0, 4), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (3, 6)] = double of [(0, 3), (0, 4), (1, 3), (1, 4), (2, 4)] along the star of 3
      Gamma edges [(0, 3), (0, 5), (1, 4), (1, 5), (2, 5), (2, 6), (3, 6), (4, 6)] = double of [(0, 2), (0, 3), (1, 3), (1, 4), (2, 4)] along the star of 0
      Gamma edges [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6), (4, 6)] = double of [(0, 2), (0, 3), (0, 4), (1, 3), (1, 4), (2, 4)] along the star of 1
      Gamma edges [(0, 3), (0, 5), (1, 4), (1, 5), (2, 5), (2, 6), (3, 6), (4, 6), (5, 6)] = double of [(0, 2), (0, 3), (0, 4), (1, 3), (1, 4), (2, 4)] along the star of 2
   of the others, obtained by gluing k >= 3 copies of such a graph along a star: 0
   named graphs on 7 vertices: not covered by Proposition 9.3? / a double along a star?
      the heptagon C_7                                                   True  / False
      the complement of C_7                                              False / False
      the hexagon with a pendant vertex                                  True  / False
      the hexagon with a vertex adjacent to two consecutive vertices     True  / False
      the hexagon with a doubled vertex (false twin)                     False / False
      the hexagon with a true twin                                       False / False
      the hexagon with a vertex adjacent to two antipodal vertices       True  / True
      the pentagon with a pendant path of length 2                       True  / False
      the pentagon with two pendant vertices at the same vertex          False / False
      the pentagon with pendant vertices at two adjacent vertices        True  / False
      the pentagon with pendant vertices at two non-adjacent vertices    True  / False
      the 4-cycle and the pentagon sharing an edge                       True  / False
      the complement of the path P_7                                     False / False
   not covered and not such a double: 91
   disconnected or with isolated vertices among them: 0
   list of the 95 graphs on 7 vertices not covered by Proposition 9.3 (edges of Gamma; the doubles are marked):
      [(0, 3), (0, 4), (1, 4), (1, 5), (2, 5), (2, 6), (3, 6)]  
      [(0, 3), (0, 5), (1, 4), (1, 6), (2, 5), (2, 6), (3, 6)]  
      [(0, 4), (0, 5), (0, 6), (1, 4), (1, 6), (2, 5), (3, 6)]  
      [(0, 4), (0, 5), (0, 6), (1, 5), (1, 6), (2, 5), (3, 6)]  
      [(0, 4), (0, 5), (1, 4), (1, 5), (2, 5), (2, 6), (3, 6)]  
      [(0, 4), (0, 5), (1, 4), (1, 6), (2, 5), (2, 6), (3, 6)]  
      [(0, 4), (0, 5), (1, 4), (2, 5), (2, 6), (3, 6), (4, 6)]  
      [(0, 4), (0, 5), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6)]  
      [(0, 4), (0, 5), (1, 5), (1, 6), (2, 5), (3, 6), (4, 6)]  
      [(0, 4), (0, 6), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 4), (1, 5), (2, 5), (2, 6), (3, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 4), (1, 5), (2, 5), (2, 6), (4, 6)]  
      [(0, 3), (0, 4), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6)]  
      [(0, 3), (0, 4), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (4, 6)]  
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (3, 6)]  
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (2, 5), (2, 6), (3, 6)]  
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (2, 5), (3, 6), (4, 6)]  
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 6), (2, 5), (2, 6), (3, 6)]  
      [(0, 3), (0, 5), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6)]  
      [(0, 3), (0, 5), (1, 4), (1, 5), (2, 5), (2, 6), (3, 6), (4, 6)]  (double along a star)
      [(0, 4), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (3, 6)]  (double along a star)
      [(0, 4), (0, 5), (0, 6), (1, 4), (1, 5), (2, 5), (2, 6), (3, 6)]  
      [(0, 4), (0, 5), (0, 6), (1, 4), (1, 5), (2, 5), (3, 6), (4, 6)]  
      [(0, 4), (0, 5), (0, 6), (1, 4), (1, 6), (2, 5), (2, 6), (3, 6)]  
      [(0, 4), (0, 5), (0, 6), (1, 4), (2, 5), (2, 6), (3, 6), (4, 6)]  
      [(0, 4), (0, 5), (1, 4), (1, 6), (2, 5), (2, 6), (3, 5), (4, 6)]  
      [(0, 4), (0, 5), (1, 4), (1, 6), (2, 5), (2, 6), (3, 6), (4, 6)]  
      [(0, 4), (0, 5), (1, 4), (1, 6), (2, 5), (3, 6), (4, 6), (5, 6)]  
      [(0, 3), (0, 4), (0, 5), (0, 6), (1, 4), (1, 5), (2, 5), (2, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 4), (1, 5), (1, 6), (2, 6), (3, 5), (4, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 4), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 4), (1, 6), (2, 5), (2, 6), (3, 5), (4, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (5, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (3, 6), (5, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 4), (1, 5), (1, 6), (2, 6), (3, 5), (4, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 4), (1, 5), (2, 5), (2, 6), (3, 5), (4, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 4), (1, 5), (2, 5), (2, 6), (3, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 4), (1, 5), (2, 5), (2, 6), (3, 6), (5, 6)]  
      [(0, 3), (0, 4), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6), (4, 6)]  
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (3, 6)]  
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6)]  
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (3, 6), (4, 6)]  
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6), (2, 6), (3, 5), (4, 6)]  
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (2, 5), (2, 6), (3, 6), (4, 6)]  
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (2, 5), (3, 6), (4, 6), (5, 6)]  
      [(0, 3), (0, 5), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6), (5, 6)]  
      [(0, 3), (0, 5), (1, 4), (1, 5), (2, 5), (2, 6), (3, 6), (4, 6), (5, 6)]  (double along a star)
      [(0, 4), (0, 5), (0, 6), (1, 4), (1, 5), (2, 4), (2, 6), (3, 5), (3, 6)]  
      [(0, 4), (0, 5), (0, 6), (1, 4), (1, 6), (2, 5), (2, 6), (3, 5), (4, 6)]  
      [(0, 4), (0, 5), (0, 6), (1, 4), (1, 6), (2, 5), (2, 6), (3, 6), (4, 6)]  
      [(0, 4), (0, 5), (1, 4), (1, 6), (2, 5), (2, 6), (3, 6), (4, 6), (5, 6)]  
      [(0, 3), (0, 4), (0, 5), (0, 6), (1, 4), (1, 5), (2, 4), (2, 6), (3, 5), (3, 6)]  
      [(0, 3), (0, 4), (0, 5), (0, 6), (1, 4), (1, 6), (2, 5), (2, 6), (3, 5), (4, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 3), (1, 4), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 3), (1, 4), (1, 6), (2, 5), (2, 6), (3, 5), (4, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 4), (1, 5), (1, 6), (2, 4), (2, 6), (3, 5), (3, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 4), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 4), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6), (5, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 3), (1, 5), (1, 6), (2, 4), (2, 5), (3, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (4, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6), (5, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (3, 5), (3, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 4), (1, 5), (1, 6), (2, 6), (3, 5), (3, 6), (4, 6)]  
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (3, 6), (4, 6)]  
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (3, 6), (5, 6)]  
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6), (2, 4), (2, 6), (3, 5), (4, 6)]  
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (4, 6)]  
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 6), (4, 6)]  (double along a star)
      [(0, 3), (0, 5), (0, 6), (1, 4), (1, 5), (2, 5), (2, 6), (3, 6), (4, 6), (5, 6)]  
      [(0, 3), (0, 5), (1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (4, 6), (5, 6)]  
      [(0, 3), (0, 4), (0, 5), (0, 6), (1, 3), (1, 4), (1, 6), (2, 5), (3, 5), (3, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 5), (0, 6), (1, 3), (1, 5), (1, 6), (2, 4), (2, 5), (3, 6), (5, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 3), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 3), (1, 5), (1, 6), (2, 4), (2, 6), (3, 5), (3, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 4), (1, 5), (1, 6), (2, 4), (2, 6), (3, 5), (3, 6), (5, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 4), (1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 3), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 5), (4, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 3), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 6), (1, 3), (1, 5), (1, 6), (2, 4), (2, 5), (3, 5), (3, 6), (4, 6)]  
      [(0, 3), (0, 5), (1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 6), (4, 5), (4, 6)]  
      [(0, 2), (0, 4), (0, 5), (0, 6), (1, 3), (1, 4), (1, 5), (2, 4), (2, 6), (3, 5), (3, 6), (4, 6)]  
      [(0, 2), (0, 4), (0, 5), (0, 6), (1, 3), (1, 5), (2, 4), (2, 5), (2, 6), (3, 6), (4, 5), (4, 6)]  
      [(0, 2), (0, 4), (0, 5), (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), (2, 6), (3, 5), (3, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 5), (0, 6), (1, 3), (1, 5), (1, 6), (2, 4), (2, 5), (3, 5), (3, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 5), (0, 6), (1, 3), (1, 5), (1, 6), (2, 4), (2, 6), (3, 5), (3, 6), (4, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 3), (1, 4), (1, 6), (2, 4), (2, 5), (2, 6), (3, 5), (3, 6), (5, 6)]  
      [(0, 3), (0, 4), (0, 5), (1, 3), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 5), (3, 6), (4, 6)]  
      [(0, 2), (0, 3), (0, 5), (0, 6), (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (3, 5), (3, 6), (4, 6)]  
      [(0, 2), (0, 4), (0, 5), (0, 6), (1, 3), (1, 4), (1, 6), (2, 4), (2, 5), (3, 5), (3, 6), (4, 6), (5, 6)]  
