  We study a family of generalizations of Edge Dominating Set on directed
graphs called Directed $(p,q)$-Edge Dominating Set. In this problem an arc
$(u,v)$ is said to dominate itself, as well as all arcs which are at distance
at most $q$ from $v$, or at distance at most $p$ to $u$.
  First, we give significantly improved FPT algorithms for the two most
important cases of the problem, $(0,1)$-dEDS and $(1,1)$-dEDS (that correspond
to versions of Dominating Set on line graphs), as well as polynomial kernels.
We also improve the best-known approximation for these cases from logarithmic
to constant. In addition, we show that $(p,q)$-dEDS is FPT parameterized by
$p+q+tw$, but W-hard parameterized by $tw$ (even if the size of the optimal is
added as a second parameter), where $tw$ is the treewidth of the underlying
graph of the input.
  We then go on to focus on the complexity of the problem on tournaments. Here,
we provide a complete classification for every possible fixed value of $p,q$,
which shows that the problem exhibits a surprising behavior, including cases
which are in P; cases which are solvable in quasi-polynomial time but not in P;
and a single case $(p=q=1)$ which is NP-hard (under randomized reductions) and
cannot be solved in sub-exponential time, under standard assumptions.
