neo4j - How to write a Cypher query to check commutativity between nodes -

i need check commutativity between nodes. there 3 relationships across node tracked following: aabbca, aabcba, aacbba. seen every b , c node placed commute. below use query on how generate graph.

here graph:

create (x1:node {title:'a'})  create (x2:node {title:'a'})  create (x3:node {title:'b'})  create (x4:node {title:'b'})  create (x5:node {title:'c'})  create (x6:node {title:'a'})  create (x1) - [:transition {id:[1]}] -> (x2)  create (x2) - [:transition {id:[1]}] -> (x3)  create (x3) - [:transition {id:[1]}] -> (x4)  create (x4) - [:transition {id:[1]}] -> (x5)  create (x5) - [:transition {id:[1]}] -> (x6)  create (x1) - [:transition {id:[2]}] -> (x2)  create (x2) - [:transition {id:[2]}] -> (x3)  create (x3) - [:transition {id:[2]}] -> (x5)  create (x5) - [:transition {id:[2]}] -> (x4)  create (x4) - [:transition {id:[2]}] -> (x6)  create (x1) - [:transition {id:[3]}] -> (x2)  create (x2) - [:transition {id:[3]}] -> (x5)  create (x5) - [:transition {id:[3]}] -> (x3)  create (x3) - [:transition {id:[3]}] -> (x4)  create (x4) - [:transition {id:[3]}] -> (x6) 

that generates this:

now identify commutative nodes, go ahead , filter on basis of relationship id 1 & 2, 1 & 3 , 2 & 3 using following query respective graphs:

match (a)-[r]->(b) 1 in r.id or 2 in r.id  return a,r,b  match (a)-[r]->(b) 1 in r.id or 3 in r.id  return a,r,b match (a)-[r]->(b) 2 in r.id or 3 in r.id return a,r,b 

case 1: case 2:: case 3:

the circled part of case 2, our example has 2 nodes, in generic case can have 1 (like see in case 1 & case 3) n number of nodes, n finite number.

now that's been said, want create cypher query check if such commutable node exists or not in graph , based on need fetch nodes , relationships graph using query this:

match (a)-[r]->(m)-[r]->(k)-[r]->[b],  (b)-[s]->(k)-[s]->(m)-[s]->(a) r.id <> s.id  , m returning minimum 1 node , k returning minimum 1 node return a,r,m,k,s,b  k = m   m = match (o)-[r]->(b1), (n)-[r]->(p)->[s]->(n)   (b1 = b return b) or  (n = m , p = m , n returning @ least 1 node , p returning @ least 1 node  return n,r,p,s) 

at point might not have defined constraints need represent above query correctly. can help?

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update: solved case 1 & case 3 (which same type of condition), not generic solution. in generic solution case 2 have worked well.:

match (a)-[r]->(b)-[s]->(c)-[t]->(d) (1 in r.id or 2 in r.id)  , (1 in s.id or 2 in s.id)  , (1 in t.id or 2 in t.id) a,r,b,s,c,t,d match (a)-[r1]->(c)-[s1]->(b)-[t1]->(d) ( r <> r1 , (1 in r1.id or 2 in r1.id)) , (s<>s1 , (1 in s1.id or 2 in s1.id)) , (t<>t1 , (1 in t1.id or 2 in t1.id))  return a,r,r1,b,s,s1,c,t,t1,d  match (a)-[r]->(b)-[s]->(c)-[t]->(d) (3 in r.id or 2 in r.id)  , (3 in s.id or 2 in s.id)  , (3 in t.id or 2 in t.id) a,r,b,s,c,t,d match (a)-[r1]->(c)-[s1]->(b)-[t1]->(d) ( r <> r1 , (3 in r1.id or 2 in r1.id)) , (s<>s1 , (3 in s1.id or 2 in s1.id)) , (t<>t1 , (3 in t1.id or 2 in t1.id))  return a,r,r1,b,s,s1,c,t,t1,d 

// nested loop through ids: unwind range(1,3) i1 i1     unwind range(i1+1,3) i2 i1, i2  // take pathways each pair of ids: match p1 = (n1:node)-[:transition*5 {id:[i1]}]->(n2:node),        p2 = (n1:node)-[:transition*5 {id:[i2]}]->(n2:node)  // take nodes of each path: i1, i2,       nodes(p1) p1,       nodes(p2) p2  i1, i2, p1, p2,       // check whether nodes different:      reduce(acc = [], x in range(0, size(p1)-1) |              acc + [ { i:x,                        is: (p1[x] <> p2[x]) ,                            (p1[x]['title'] <> p2[x]['title'])                      } ]      ) iscomm  i1, i2, p1, p2,      // take nodes different:       filter(pair in iscomm pair['is'] = true) iscomm  return i1 id1, i2 id2,        // convert map array:        reduce(acc = [], pair in iscomm |                acc + [ [ p1[pair['i']],                          p2[pair['i']]                     ] ]         ) commutative