3 research outputs found
On Directed Covering and Domination Problems
In this paper, we study covering and domination problems on directed graphs.
Although undirected Vertex Cover and Edge Dominating Set are well-studied classical graph problems, the directed versions have not been studied much due to the lack of clear definitions.
We give natural definitions for Directed r-In (Out) Vertex Cover and Directed (p,q)-Edge Dominating Set as directed generations of Vertex Cover and Edge Dominating Set.
For these problems, we show that
(1) Directed r-In (Out) Vertex Cover and Directed (p,q)-Edge Dominating Set are NP-complete on planar directed acyclic graphs except when r=1 or (p,q)=(0,0),
(2) if r>=2, Directed r-In (Out) Vertex Cover is W[2]-hard and (c*ln k)-inapproximable on directed acyclic graphs,
(3) if either p or q is greater than 1, Directed (p,q)-Edge Dominating Set is W[2]-hard and (c*ln k)-inapproximable on directed acyclic graphs,
(4) all problems can be solved in polynomial time on trees, and
(5) Directed (0,1),(1,0),(1,1)-Edge Dominating Set are fixed-parameter tractable in general graphs.
The first result implies that (directed) r-Dominating Set on directed line graphs is NP-complete even if r=1
New Results on Directed Edge Dominating Set
We study a family of generalizations of Edge Dominating Set on directed
graphs called Directed -Edge Dominating Set. In this problem an arc
is said to dominate itself, as well as all arcs which are at distance
at most from , or at distance at most to .
First, we give significantly improved FPT algorithms for the two most
important cases of the problem, -dEDS and -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 -dEDS is FPT parameterized by
, but W-hard parameterized by (even if the size of the optimal is
added as a second parameter), where 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 ,
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 which is NP-hard (under randomized reductions) and
cannot be solved in sub-exponential time, under standard assumptions