  The subset sum problem is one of the simplest and most fundamental NP-hard
problems in combinatorial optimization. We consider two extensions of this
problem: The subset sum problem with digraph constraint (SSG) and subset sum
problem with weak digraph constraint (SSGW). In both problems there is given a
digraph with sizes assigned to the vertices. Within SSG we want to find a
subset of vertices whose total size does not exceed a given capacity and which
contains a vertex if at least one of its predecessors is part of the solution.
Within SSGW we want to find a subset of vertices whose total size does not
exceed a given capacity and which contains a vertex if all its predecessors are
part of the solution. SSG and SSGW have been introduced recently by Gourves et
al. who studied their complexity for directed acyclic graphs and oriented
trees. We show that both problems are NP-hard even on oriented co-graphs and
minimal series-parallel digraphs. Further, we provide pseudo-polynomial
solutions for SSG and SSGW with digraph constraints given by directed co-graphs
and series-parallel digraphs.
