Imagine I have a graph $(V,E)$ of vertices $V$ and undirected edges $E$. Then a clique is a set of vertices $(v_1,...,v_n)$ such that that there is an edge between any pair $(v_i,v_j)$ of vertices inside this set $(v_1,...,v_n)$.
The problem of a maximum clique is finding a clique which contains as much vertices as possible and the problem is NP-complete.
Now imagine I have given a directed and acyclic graph. Then I can reorder the graph in such a way that there can only be an edge from $i$ to $j$ if $i<j$ (so the corresponding adjacency matrix would be an upper triangle matrix). Now I want to find a maximum set of vertices $(v_1,...,v_n)$ such that between any pair $(v_i,v_j)$ of vertices inside the set with $i<j$ there is a directed edge from $v_i$ to $v_j$ . It is somehow the directed version of the maximum clique problem. So e.g. if a clique set is given by $(v_1,v_2,v_3)$, then there must be an edge $v_1v_2$, $v_1v_3$ and $v_2v_3$.
Now I am of course wondering if this problem would still be NP-complete... Does anyone have an idea?
I just realized a mistake.... If I just change all directed in undirected edges then it reduces to a regular maximum clique problem, right? Hence, the problem is NP-complete?