The first case
To get the whole connected graph, you need to use a wildcard property path to follow most of the path, and then grab the last link with an actual variable. I usually use the empty relative path in constructing wildcards, so as to use <>|!<>
as the wildcard, but since you mentioned that your endpoint doesn't like it, you can use any absolute IRI that you like, too. E.g.,
prefix x: <urn:ex:>
construct { ?s ?p ?o }
where { :A (x:|!x:)* ?s . ?s ?p ?o . }
This works because every property is either x: or not, so x:|!x: matches every property, and then (x:|!x:)* is an arbitrary length path, including paths of length zero, which means that ?s will be bound to everything reachable from :a, including :a itself. Then you're grabbing the triples where ?s is the subject. When you construct the graph of all those triples, you get the subgraph that you're looking for.
Here's an example based on the graph you showed. I used different properties for different edges to show that it works, but this will work if they're all the same, too.
@prefix : <urn:ex:> .
:A :p :B, :C .
:B :q :D .
:C :r :E .
:F :s :G .
:G :t :H .
prefix x: <urn:ex:>
prefix : <urn:ex:>
construct {
?s ?p ?o
}
where {
:A (x:|!x:)* ?s .
?s ?p ?o .
}
Since this is a construct query, the result is a graph, not a "table". It contains the triples we'd expect:
@prefix : <urn:ex:> .
:C :r :E .
:B :q :D .
:A :p :B , :C .
The second case
If you want to ensure that the paths end in a particular kind of edge, you can do that too. If you only want the paths from A1 to those ending with edges on d, you can do:
prefix x: <urn:ex:> #-- arbitrary, used for the property path.
prefix : <...> #-- whatever you need for your data.
construct {
?s1 ?p ?o1 . #-- internal edge in the path
?s2 :d ?o2 . #-- final edge in the path
}
where {
:A (x:|!x:)* ?s1 . #-- start at :A and go any length into the path
?s1 ?p ?o1 . #-- get the triple from within the path, but make
?o1 (x:|!x:)* ?s2 . #-- sure that from ?o1 you can get to to some other
?s2 :d ?o2 . #-- ?s2 that's related to an ?o2 by property :d .
}