graphs up to isomorphism, n=1..7: [1, 2, 4, 11, 34, 156, 1044]
brute-force cross-check of the enumeration for n<=6: OK
join / chordal predicates cross-checked against the definitions (n<=6): OK
Lemma 2.6 (definition of K vs atoms, two different decompositions), connected graphs n<=7: OK; number of graphs in K: {1: 1, 2: 1, 3: 2, 4: 6, 5: 18, 6: 75, 7: 368}
n=1: graphs 1, joins 0, not joins 1: (i) 1 (ii) 1 (iii) 1 | connected 1: (i) 1 (ii) 1 (iii) 1
n=2: graphs 2, joins 1, not joins 1: (i) 1 (ii) 1 (iii) 1 | connected 0: (i) 0 (ii) 0 (iii) 0
n=3: graphs 4, joins 2, not joins 2: (i) 2 (ii) 2 (iii) 2 | connected 0: (i) 0 (ii) 0 (iii) 0
n=4: graphs 11, joins 5, not joins 6: (i) 6 (ii) 6 (iii) 6 | connected 1: (i) 1 (ii) 1 (iii) 1
n=5: graphs 34, joins 13, not joins 21: (i) 21 (ii) 21 (iii) 21 | connected 8: (i) 8 (ii) 8 (iii) 8
n=6: graphs 156, joins 44, not joins 112: (i) 93 (ii) 105 (iii) 112 | connected 68: (i) 49 (ii) 61 (iii) 68
n=7: graphs 1044, joins 191, not joins 853: (i) 477 (ii) 578 (iii) 758 | connected 662: (i) 305 (ii) 394 (iii) 567

six vertices: graphs that are not joins and do not satisfy the hypothesis of Theorem 1.5:
  number: 7  equal to the seven graphs of Table 2: True
  2: pentagon with a pendant vertex             edges(canon) 03 04 05 12 14 23            connected=True  (d): K0 on [0, 1, 2, 3, 4, 5], twins [], complement of K0: Lambda_{m,I} with m=5, |I|=1
  1: C4 with a pendant path of length 2         edges(canon) 02 04 05 14 15 23            connected=True  (d): K0 on [0, 1, 2, 3, 4, 5], twins [], complement of K0: in T: true twin of a graph that is [in K (clique-sum of complete graphs and cycles)]
  4: C4 and a triangle with a common vertex     edges(canon) 02 03 04 05 14 15 23         connected=True  (d): K0 on [0, 1, 2, 3, 4, 5], twins [], complement of K0: in T: true twin of a graph that is [in K (clique-sum of complete graphs and cycles)]
  6: pentagon with a false twin                 edges(canon) 03 04 05 12 14 15 23         connected=True  (d): K0 on [0, 1, 2, 3, 4, 5], twins [], complement of K0: in T: true twin of a graph that is [cycle C_5]
  3: hexagon                                    edges(canon) 04 05 13 15 23 24            connected=True  (d): K0 on [0, 1, 2, 3, 4, 5], twins [], complement of K0: prism
  5: pentagon with a triangle on an edge        edges(canon) 01 04 05 13 15 23 24         connected=True  (d): K0 on [0, 1, 2, 3, 4, 5], twins [], complement of K0: Lambda_{m,I} with m=5, |I|=2
  7: pentagon with a true twin                  edges(canon) 03 04 05 12 15 23 24 34      connected=True  (d): K0 on [0, 1, 2, 3, 5], twins [(3, 4)], complement of K0: cycle C_5

seven vertices: not covered by Proposition 9.3: 95
  all connected: True  edge numbers: min 7 max 13
  of these, doubles along stars of connected non-join graphs with <= 6 vertices: 4
     03 04 06 13 14 15 23 24  <-  (5, '02 03 04 12 13', 1) (1 ways)
     04 05 06 12 13 16 25 34  <-  (5, '03 04 12 14 23', 0) (5 ways)
     01 04 05 06 12 13 16 25 34  <-  (5, '01 03 04 12 14 23', 4) (1 ways)
     02 03 04 05 06 12 13 16 25 34  <-  (5, '01 03 04 12 14 23', 2) (2 ways)
  without answer: 91

  heptagon C7                                                                      uncovered=True  double=False
  hexagon with a pendant vertex                                                    uncovered=True  double=False
  pentagon with a pendant path of length two                                       uncovered=True  double=False
  pentagon with pendant vertices at two different vertices (adjacent)              uncovered=True  double=False
  pentagon with pendant vertices at two different vertices (non-adjacent)          uncovered=True  double=False
  4-cycle and pentagon with a common edge                                          uncovered=True  double=False
  double of the pentagon along a star (two pentagons sharing a path with 3 vertices) uncovered=True  double=True
