Authors: Rémy Belmonte ; Tesshu Hanaka ; Ioannis Katsikarelis ; Eun Jung Kim ; Michael Lampis

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.

• 05C20 - Directed graphs (digraphs), tournaments
• 05C69 - Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
• 05C85 - Graph algorithms (graph-theoretic aspects)
• 68Q25 - Analysis of algorithms and problem complexity
• 68R10 - Graph theory (including graph drawing) in computer science