(1) networkx atlas agrees with my enumeration for n = 1..7: True

(2) certificates, graphs with at most 6 vertices that are not joins
   n=1: {'edgeless': 1} total 1
   n=2: {'edgeless': 1} total 1
   n=3: {'chordal': 1, 'edgeless': 1} total 2
   n=4: {'chordal': 5, 'edgeless': 1} total 6
   n=5: {'chordal': 16, 'co-K': 3, 'edgeless': 1, 'join': 1} total 21
   n=6: {'(d)': 7, 'chordal': 66, 'chordal+join': 1, 'co-K': 33, 'edgeless': 1, 'join': 4} total 112
   the 112 graphs with six vertices (edges; number of isolated vertices; kind of each component of Gamma^-):
     (no edges)                       iso=6  edgeless
     01                               iso=4  chordal
     01 02                            iso=3  chordal
     01 02 03                         iso=2  chordal
     01 02 03 04                      iso=1  chordal
     01 02 12                         iso=3  chordal
     03 12                            iso=2  chordal | chordal
     01 03 12                         iso=2  chordal
     01 02 03 12                      iso=2  chordal
     03 04 12                         iso=1  chordal | chordal
     01 03 04 12                      iso=1  chordal
     01 02 03 04 12                   iso=1  chordal
     03 04 05 12                      iso=0  chordal | chordal
     01 03 04 05 12                   iso=0  chordal
     02 03 12 13                      iso=2  join
     01 02 03 12 13                   iso=2  chordal
     02 04 12 13                      iso=1  chordal
     01 02 04 12 13                   iso=1  chordal
     02 03 04 12 13                   iso=1  co-K[K2,K2,K2,K2,K3]
     01 02 03 04 12 13                iso=1  chordal
     04 05 12 13                      iso=0  chordal | chordal
     01 04 05 12 13                   iso=0  chordal
     02 04 05 12 13                   iso=0  chordal
     01 02 04 05 12 13                iso=0  chordal
     02 03 04 05 12 13                iso=0  co-K[K2,K2,K3,K3,K3,K4]
     02 03 04 12 13 14                iso=1  join
     01 02 03 04 12 13 14             iso=1  chordal
     02 03 05 12 13 14                iso=0  co-K[C4,K3,K4]
     01 02 03 05 12 13 14             iso=0  chordal
     02 03 04 05 12 13 14             iso=0  co-K[K2,K2,K2,K2,K3,K4]
     01 02 03 12 13 23                iso=2  chordal
     04 12 13 23                      iso=1  chordal | chordal
     01 04 12 13 23                   iso=1  chordal
     01 02 04 12 13 23                iso=1  chordal
     01 02 03 04 12 13 23             iso=1  chordal
     04 05 12 13 23                   iso=0  chordal | chordal
     01 04 05 12 13 23                iso=0  chordal
     01 02 04 05 12 13 23             iso=0  chordal
     01 02 03 04 14 23                iso=1  chordal
     05 14 23                         iso=0  chordal | chordal | chordal
     01 05 14 23                      iso=0  chordal | chordal
     01 02 05 14 23                   iso=0  chordal
     02 03 05 14 23                   iso=0  chordal | chordal
     01 02 03 05 14 23                iso=0  chordal
     03 04 12 14 23                   iso=1  co-K[C5]
     01 03 04 12 14 23                iso=1  co-K[K2,K2,K2,K2,K2,K2]
     01 02 03 04 12 14 23             iso=1  chordal
     01 02 05 12 14 23                iso=0  chordal
     03 05 12 14 23                   iso=0  chordal
     01 03 05 12 14 23                iso=0  co-K[K3,K3,K3,K3,K3,K3]
     02 03 05 12 14 23                iso=0  chordal
     01 02 03 05 12 14 23             iso=0  chordal
     03 04 05 12 14 23                iso=0  (d): Lambda_{m,I} with m=5, |I|=1
     01 03 04 05 12 14 23             iso=0  co-K[K2,K2,K3,K3,K3,K3,K3]
     02 03 04 12 13 14 23             iso=1  join
     01 02 03 04 12 13 14 23          iso=1  chordal
     02 03 05 12 13 14 23             iso=0  chordal
     01 02 03 05 12 13 14 23          iso=0  chordal
     04 05 12 13 14 23                iso=0  chordal
     01 04 05 12 13 14 23             iso=0  chordal
     02 04 05 12 13 14 23             iso=0  co-K[C4,K3,K3,K3]
     01 02 04 05 12 13 14 23          iso=0  chordal
     02 03 04 05 12 13 14 23          iso=0  co-K[K2,K2,K2,K2,K3,K3,K3]
     04 05 14 15 23                   iso=0  join | chordal
     01 04 05 14 15 23                iso=0  chordal | chordal
     02 04 05 14 15 23                iso=0  (d): in T: true twin of a graph that is [in K (clique-sum of complete graphs and cycles)]
     01 02 04 05 14 15 23             iso=0  chordal
     02 03 04 05 14 15 23             iso=0  (d): in T: true twin of a graph that is [in K (clique-sum of complete graphs and cycles)]
     02 04 05 12 14 15 23             iso=0  co-K[K2,K2,K3,K3,K3,K3]
     01 02 04 05 12 14 15 23          iso=0  chordal
     03 04 05 12 14 15 23             iso=0  (d): in T: true twin of a graph that is [cycle C_5]
     01 03 04 05 12 14 15 23          iso=0  co-K[K2,K2,K2,K2,K3,K3,K3,K3]
     02 03 04 05 12 14 15 23          iso=0  co-K[K2,K2,K2,K2,K3,K3,K3]
     01 02 03 04 13 14 23 24          iso=1  join
     01 03 05 13 14 23 24             iso=0  co-K[C4,K3,K3]
     01 02 03 05 13 14 23 24          iso=0  co-K[K2,K2,K2,K2,K3,K3,K3]
     01 02 03 04 12 13 14 23 24       iso=1  chordal
     03 05 12 13 14 23 24             iso=0  chordal
     01 03 05 12 13 14 23 24          iso=0  chordal
     01 02 03 05 12 13 14 23 24       iso=0  chordal
     03 04 05 12 13 14 23 24          iso=0  co-K[C4,K2,K2,K3]
     01 03 04 05 12 13 14 23 24       iso=0  co-K[K2,K2,K2,K2,K2,K2,K3,K3]
     04 05 13 15 23 24                iso=0  (d): prism
     01 04 05 13 15 23 24             iso=0  (d): Lambda_{m,I} with m=5, |I|=2
     01 02 04 05 13 15 23 24          iso=0  co-K[C4,K2,K2,K3,K3]
     03 04 05 13 15 23 24             iso=0  co-K[C4,K3,K3]
     01 03 04 05 13 15 23 24          iso=0  co-K[C4,K2,K2,K3]
     01 02 04 05 12 13 15 23 24       iso=0  chordal
     03 04 05 12 13 15 23 24          iso=0  co-K[C5,K3]
     01 03 04 05 12 13 15 23 24       iso=0  co-K[K2,K2,K2,K2,K2,K2,K3,K3]
     03 04 05 13 14 15 23 24          iso=0  co-K[K2,K2,K2,K2,K3,K3]
     01 03 04 05 13 14 15 23 24       iso=0  co-K[K2,K2,K2,K2,K2,K2,K3]
     02 03 04 05 13 14 15 23 24       iso=0  co-K[K2,K2,K2,K2,K2,K2,K3]
     01 02 03 04 12 13 14 23 24 34    iso=1  chordal
     05 12 13 14 23 24 34             iso=0  chordal | chordal
     01 05 12 13 14 23 24 34          iso=0  chordal
     01 02 05 12 13 14 23 24 34       iso=0  chordal
     01 02 03 05 12 13 14 23 24 34    iso=0  chordal
     01 05 15 23 24 34                iso=0  chordal | chordal
     01 02 05 15 23 24 34             iso=0  chordal
     01 02 03 05 15 23 24 34          iso=0  chordal
     01 03 05 12 15 23 24 34          iso=0  co-K[C4,C4]
     01 02 03 05 12 15 23 24 34       iso=0  chordal
     03 04 05 12 15 23 24 34          iso=0  (d): cycle C_5 + 1 true twin(s)
     01 03 04 05 12 15 23 24 34       iso=0  co-K[C4,K2,K2,K2,K2]
     02 03 04 05 12 15 23 24 34       iso=0  co-K[C4,K2,K2,K2,K2]
     01 02 03 05 12 13 15 23 24 34    iso=0  chordal
     02 04 05 12 13 15 23 24 34       iso=0  co-K[C5,K2,K2]
     01 02 04 05 12 13 15 23 24 34    iso=0  co-K[K2,K2,K2,K2,K2,K2,K2,K2]
     02 03 04 05 12 13 15 23 24 34    iso=0  co-K[K2,K2,K2,K2,K2,K2,K2,K2]
     03 04 05 12 14 15 23 25 34       iso=0  co-K[C6]
     01 03 04 05 12 14 15 23 25 34    iso=0  co-K[K2,K2,K2,K2,K2,K2,K2,K2]

(3) my list of the 95 graphs with seven vertices that do not satisfy the hypothesis of Proposition 9.3
     02 04 05 06 14 15 23                      
     02 04 06 12 14 15 23                      
     03 04 06 12 14 15 23                      
     04 05 06 12 14 15 23                      
     01 04 06 13 15 23 24                      
     01 02 04 06 13 15 23 24                   
     03 04 06 13 15 23 24                      
     01 03 04 06 13 15 23 24                   
     04 05 06 13 15 23 24                      
     01 04 05 06 13 15 23 24                   
     01 02 04 05 06 13 15 23 24                
     03 04 05 06 13 15 23 24                   
     01 03 04 05 06 13 15 23 24                
     03 04 06 12 13 15 23 24                   
     04 05 06 12 13 15 23 24                   
     03 04 06 13 14 15 23 24                     (double along a star)
     05 06 13 14 15 23 24                      
     03 05 06 13 14 15 23 24                   
     03 05 06 12 13 14 15 23 24                
     03 05 06 13 14 16 23 24 25                
     01 03 06 12 15 23 24 34                   
     03 05 06 12 15 23 24 34                   
     01 03 05 06 12 15 23 24 34                
     02 03 05 06 12 15 23 24 34                
     02 05 06 12 13 15 23 24 34                
     02 03 06 12 13 14 15 25 34                
     03 05 06 12 16 25 34                      
     01 03 05 06 12 16 25 34                   
     02 04 06 12 13 16 25 34                   
     01 02 04 06 12 13 16 25 34                
     03 04 06 12 13 16 25 34                   
     02 03 04 06 12 13 16 25 34                
     04 05 06 12 13 16 25 34                     (double along a star)
     01 04 05 06 12 13 16 25 34                  (double along a star)
     02 04 05 06 12 13 16 25 34                
     01 02 04 05 06 12 13 16 25 34             
     02 03 04 05 06 12 13 16 25 34               (double along a star)
     02 03 05 06 12 13 14 16 25 34             
     02 03 06 14 15 23 25 34                   
     01 02 03 06 14 15 23 25 34                
     01 03 04 06 14 15 23 25 34                
     01 02 03 04 06 14 15 23 25 34             
     02 03 06 12 14 15 23 25 34                
     03 04 06 12 14 15 23 25 34                
     05 06 12 14 15 23 25 34                   
     01 05 06 12 14 15 23 25 34                
     03 05 06 12 14 15 23 25 34                
     04 05 06 12 13 14 15 23 25 34             
     04 05 06 12 13 16 23 25 34                
     05 06 14 16 23 25 34                      
     01 05 06 14 16 23 25 34                   
     01 02 05 06 14 16 23 25 34                
     03 05 06 14 16 23 25 34                   
     01 03 05 06 14 16 23 25 34                
     02 03 05 06 14 16 23 25 34                
     01 02 03 05 06 14 16 23 25 34             
     03 04 05 06 14 16 23 25 34                
     01 03 04 05 06 14 16 23 25 34             
     03 05 06 12 14 16 23 25 34                
     03 04 05 06 12 14 16 23 25 34             
     02 05 06 13 14 16 23 25 34                
     02 03 05 06 13 14 16 23 25 34             
     04 05 06 13 14 16 23 25 34                
     01 04 05 06 13 14 16 23 25 34             
     02 04 05 06 13 14 16 23 25 34             
     04 05 06 12 13 14 16 23 25 34             
     03 04 05 06 12 14 15 16 23 25 34          
     03 05 06 13 14 16 23 24 25 34             
     03 05 06 15 16 23 24 25 34                
     01 03 05 06 15 16 23 24 25 34             
     02 03 05 06 15 16 23 24 25 34             
     03 05 06 12 15 16 23 24 25 34             
     04 05 06 13 15 16 23 24 25 34             
     02 04 05 06 13 15 16 23 24 25 34          
     01 03 05 06 12 15 16 25 26 34             
     02 04 05 06 12 13 15 16 25 26 34          
     02 04 06 12 13 14 15 24 25 34 35          
     03 04 06 12 14 16 24 25 34 35             
     03 05 06 12 14 16 24 25 34 35             
     02 05 06 14 16 23 24 25 34 35             
     01 02 05 06 14 16 23 24 25 34 35          
     02 04 05 06 14 16 23 24 25 34 35          
     01 02 04 05 06 14 16 23 24 25 34 35       
     02 05 06 12 14 16 23 24 25 34 35          
     03 05 06 12 14 16 23 24 25 34 35          
     03 04 05 06 12 14 16 23 24 25 34 35       
     03 04 05 06 12 15 16 24 26 34 35          
     04 05 06 12 13 15 16 24 26 34 35          
     02 04 05 06 12 13 15 16 24 26 34 35       
     04 05 06 12 13 15 16 23 24 26 34 35       
     03 04 05 06 12 16 23 24 25 34 35 45       
     01 02 05 06 12 14 16 23 26 34 35 45       
     02 03 05 06 12 14 16 23 26 34 35 45       
     02 03 05 06 12 13 14 16 23 26 34 35 45    
     01 04 05 06 13 15 16 23 24 26 34 35 45    

(4) connected graphs with at most 5 vertices that are not chordal
     n=4 edges 02 03 12 13                join=True  complement edges 01 23            complement connected=False chordal=True in K=False
     n=5 edges 02 03 04 12 13             join=False  complement edges 01 14 23 24 34   complement connected=True chordal=True in K=True
     n=5 edges 02 03 04 12 13 14          join=True  complement edges 01 23 24 34      complement connected=False chordal=True in K=False
     n=5 edges 03 04 12 14 23             join=False  complement edges 01 02 13 24 34   complement connected=True chordal=False in K=True
     n=5 edges 01 03 04 12 14 23          join=False  complement edges 02 13 24 34      complement connected=True chordal=True in K=True
     n=5 edges 02 03 04 12 13 14 23       join=True  complement edges 01 24 34         complement connected=False chordal=True in K=False
     n=5 edges 01 02 03 04 13 14 23 24    join=True  complement edges 12 34            complement connected=False chordal=True in K=False
