Find whether graph has a cycle - sql

Question

I want to find out whether it is possible to find cycles in Hierarchical or Chain data with SQL.
E.g. I have following schema:
http://sqlfiddle.com/#!3/27269
create table node (
id INTEGER
);
create table edges (
id INTEGER,
node_a INTEGER,
node_b INTEGER
);
create table graph (
id INTEGER,
edge_id INTEGER);
INSERT INTO node VALUES (1) , (2), (3), (4);
INSERT INTO edges VALUES (1, 1, 2), (2, 2, 3) , (3, 3, 4) , (4, 4, 1);
-- first graph [id = 1] with cycle (1 -> 2 -> 3 -> 4 -> 1)
INSERT INTO graph VALUES (1, 1), (1, 2), (1, 3), (1, 4);
-- second graph [id =2] without cycle (1 -> 2 -> 3)
INSERT INTO graph VALUES (2, 1), (2, 2), (2, 3);
In graph table records with same ID belong to one graph.
I need a query that will return IDs of all graphs that have at least one cycle.
So for example above query should return 1, which is the id of the first graph;

Answer 1

First, I assume this is a directed graph. An undirected graph has a trivial cycle if it contains a single edge.
The only tricky part to the recursive CTE is stopping when you've hit a cycle -- so you don't get infinite recursion.
Try this:
with cte as (
select e.object_a, e.object_b, iscycle = 0
from edges e
union all
select cte.object_a, e.object_b,
(case when cte.object_a = e.object_b then 1 else 0 end) as iscycle
from cte join
edges e
on cte.object_b = e.object_a
where iscycle = 0
)
select max(iscycle)
from cte;

Answer 2

I wrote SQL query based on #gordon-linoff answer. In some cases I had infinite loop, so I added column with node_path and then I was checking if the current connection had appeared in that column.
This is this script:
create table edges (
node_a varchar(20),
node_b varchar(20)
);
INSERT INTO edges VALUES ('A', 'B'), ('A', 'C'), ('A', 'D'), ('B', 'D'), ('D', 'K'), ('K', 'A')
GO
with cte as (
SELECT
e.node_a
, e.node_b
, 0 as depth
, iscycle = 0
, CAST(e.node_a +' -> '+ e.node_b AS varchar(MAX)) as nodes_path
FROM edges e
UNION ALL
SELECT
cte.node_a
, e.node_b
, depth + 1
, (case when cte.node_a = e.node_b then 1 else 0 end) as iscycle
, CAST(cte.nodes_path+' -> '+ e.node_b AS varchar(MAX)) as nodes_path
FROM cte
JOIN edges e ON cte.node_b = e.node_a AND cte.nodes_path NOT LIKE '%' + CAST(cte.node_a+' -> '+ e.node_b AS varchar(500)) + '%'
where iscycle = 0
)
SELECT * -- max(iscycle)
FROM cte
option (maxrecursion 300) --just for safety :)
I don't know if it is efficient where are millions of records, so if you can see that I could write this query more optimized, please share with your opinion.