Express abscence of edges in Cypher - neo4j

Question

how do I express the following in Cypher
"Return all nodes with at least one incoming edge of type A and no outgoing edges".
Best Regards

Answer 1

You can use a pattern to exclude nodes from the result subset like this:
MATCH ()-[:A]->(n) WHERE NOT (n)-->() RETURN n

Answer 2

Try
MATCH (n)
WHERE ()-[:A]->n AND NOT n-->()
RETURN n
or
MATCH ()-[:A]->(n)
WHERE NOT n-->()
RETURN DISTINCT n
Edit
Pattern expressions can be used both for pattern matching and as predicates for filtering. If used in the MATCH clause, the paths that answer the pattern are included in the result. If used for filtering, in the WHERE clause, the pattern serves as a limiting condition on the paths that have previously been matched. The result is limited, not extended to include the filter condition. When a pattern is used as a predicate for filtering, the negation of that predicate is also a predicate that can be used as a filter condition. No path answers to the negation of a pattern (if there is such a thing) so negations of patterns cannot be used in the MATCH clause. The phrase
Return all nodes with at least one incoming edge of type A and no outgoing edges
involves two patterns on nodes n, namely any incoming relationship [:A] on n and any outgoing relationship on n. The second must be interpreted as a pattern for a predicate filter condition since it involves a negation, not any outgoing relationship on n. The first, however, can be interpreted either as a pattern to match along with n, or as another pattern predicate filter condition.
These two interpretations give rise to the two cypher queries above. The first query matches all nodes and uses both patterns to filter the result. The second matches the incoming relationship on n along with n and uses the second pattern to filter the results.
The first query will match every node only once before the filtering happens. It will therefore return one result item per node that meets the criteria. The second query will match the pattern any incoming relationship [:A] on n once for each path, i.e. once for each incoming relationship on n. It may therefore contain a node multiple times in the result, hence the DISTINCT keyword to remove doubles.
If the items of interest are precisely the nodes, then using both patterns for predicates in the WHERE clause seems to me the correct interpretation. It is also more efficient since it needs to find only zero or one incoming [:A] on n to resolve the predicate. If the incoming relationships are also of interest, then some version of the second query is the right choice. One would need to bind the relationship and do something useful with it, such as return it.
Below are the execution plans for the two queries executed on a 'fresh' neo4j console.
First query:
----
Filter
|
+AllNodes
+----------+------+--------+-------------+------------------------------------------------------------------------------------------------------------------------------+
| Operator | Rows | DbHits | Identifiers | Other |
+----------+------+--------+-------------+------------------------------------------------------------------------------------------------------------------------------+
| Filter | 0 | 0 | | (nonEmpty(PathExpression((17)-[ UNNAMED18:A]->(n), true)) AND NOT(nonEmpty(PathExpression((n)-[ UNNAMED36]->(40), true)))) |
| AllNodes | 6 | 7 | n, n | |
+----------+------+--------+-------------+------------------------------------------------------------------------------------------------------------------------------+
Second query:
----
Distinct
|
+Filter
|
+TraversalMatcher
+------------------+------+--------+-------------+--------------------------------------------------------------+
| Operator | Rows | DbHits | Identifiers | Other |
+------------------+------+--------+-------------+--------------------------------------------------------------+
| Distinct | 0 | 0 | | |
| Filter | 0 | 0 | | NOT(nonEmpty(PathExpression((n)-[ UNNAMED30]->(34), true))) |
| TraversalMatcher | 0 | 13 | | n, UNNAMED8, n |
+------------------+------+--------+-------------+--------------------------------------------------------------+