1. graphs up to isomorphism, n = 1..7 : [1, 2, 4, 11, 34, 156, 1044]
   cross-check for n <= 6 against the canonical forms of all labelled graphs : agrees
   "join" (definition) = "complement disconnected" for n <= 6, and the two chordality tests for n <= 7 : agree
   not joins                           : [1, 1, 2, 6, 21, 112, 853]
   connected and not joins             : [1, 0, 0, 1, 8, 68, 662]
2. Lemma 2.6 (class K by the definition = by atoms) for all connected graphs, n <= 7 : agrees ; connected graphs in K : [1, 1, 2, 6, 18, 75, 368]
3. Table 1 (own numbers)
   n | not joins: all (i) (ii) (iii) | connected: all (i) (ii) (iii)
   1 |    1    1    1    1 |    1    1    1    1
   2 |    1    1    1    1 |    0    0    0    0
   3 |    2    2    2    2 |    0    0    0    0
   4 |    6    6    6    6 |    1    1    1    1
   5 |   21   21   21   21 |    8    8    8    8
   6 |  112   93  105  112 |   68   49   61   68
   7 |  853  477  578  758 |  662  305  394  567
   agreement with the printed Table 1 (rows 4..7): True
   n = 1, 2, 3: not joins [1, 1, 2], all satisfying the hypothesis of Theorem 1.5: True
4. six vertices: graphs that are not joins and do not satisfy the hypothesis of Theorem 1.5
   number: 7 ; all connected: True ; all satisfy Proposition 9.3: True
   the set equals the seven edge lists of Table 2 (read from main.tex): True
   drawing no. k of Figure 5 is isomorphic to row k of Table 2, k = 1..7: True
   row 1: 04 05 13 14 23 24          kinds of the graph: d ; false twins [[1, 2]]
   row 2: 01 04 12 15 23 34          kinds of the graph: d ; complement is Lambda_{5,I} with |I| = 1
   row 3: 01 05 12 23 34 45          kinds of the graph: d ; hexagon
   row 4: 02 05 13 14 23 34 35       kinds of the graph: d ; false twins [[2, 5]]; true twins [[1, 4]]
   row 5: 02 05 12 13 23 34 45       kinds of the graph: d ; complement is Lambda_{5,I} with |I| = 2
   row 6: 01 04 05 12 23 34 35       kinds of the graph: d ; false twins [[4, 5]]
   row 7: 01 04 05 12 15 23 25 34    kinds of the graph: d ; true twins [[1, 5]]
5. seven vertices
   not joins: 853 ; hypothesis of Theorem 1.5: 578 ; of Proposition 9.3: 758 ; not: 95
   the 95 graphs are all connected: True ; numbers of edges: [7, 8, 9, 10, 11, 12, 13]
   by number of edges: [(7, 10), (8, 18), (9, 27), (10, 21), (11, 10), (12, 7), (13, 2)]
   doubles along stars (7 vertices, connected, not joins) of connected non-join graphs with <= 6 vertices: 57 graphs; 4 of them among the 95
      03 06 12 16 25 35 45 46        = double of [03 04 12 14 23] along the star of 0 (5 vertices)
      04 13 25 26 35 36 45 46        = double of [04 12 13 24 34] along the star of 1 (5 vertices)
      03 06 15 16 24 26 34 35 46 56  = double of [01 04 13 23 24 34] along the star of 0 (5 vertices)
      03 06 12 16 25 35 45 46 56     = double of [01 04 13 23 24 34] along the star of 2 (5 vertices)
   without answer: 91
   named graphs:
      heptagon C_7                                                                among the 95: True  double: False
      hexagon with a pendant vertex                                               among the 95: True  double: False
      pentagon with a pendant path of length two                                  among the 95: True  double: False
      pentagon with pendant vertices at two adjacent vertices                     among the 95: True  double: False
      pentagon with pendant vertices at two non-adjacent vertices                 among the 95: True  double: False
      4-cycle and pentagon with a common edge                                     among the 95: True  double: False
      two pentagons sharing a path with three vertices (double of the pentagon)   among the 95: True  double: True
   the double of the pentagon along a star is the last named graph: True
6. Theorem 1.5 together with Theorem 1.6(1)-(3) as stated, without Proposition 9.3 (point F1)
   n = 6: 112 of 112 graphs (68 of the 68 connected ones)
   n = 7: 710 of 853 graphs (526 of the 662 connected ones)
   graphs with seven vertices that satisfy Proposition 9.3 but neither theorem as stated: 48 (41 connected)
   example of the third run, edges 02 03 04 06 12 14 15 23: Theorem 1.5: False ; Theorem 1.6(1)-(3): False ; Proposition 9.3: True ; complement in T: True, in K: False
7. connected graphs without clique separator that are neither complete nor cycles (point F2)
   n = 4: 0 graphs, 0 of them with connected complement
   n = 5: 3 graphs, 0 of them with connected complement
      edges 03 04 13 14 23 24                complement: 01 02 12 34
      edges 02 03 04 12 13 14 24 34          complement: 01 23
      edges 01 02 13 14 23 24 34             complement: 03 04 12
   n = 6: 22 graphs, 8 of them with connected complement
   K_{2,3}: clique separator: False ; complement chordal: True ; complement connected: False
   prism: clique separator: False ; complete or cycle: False
