Abstract
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 Directedr-In (Out) Vertex Cover and Directed(p,q)-Edge Dominating Set as directed generalizations of Vertex Cover and Edge Dominating Set. For these problems, we show that (1) Directedr-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, Directedr-In (Out) Vertex Cover is W[2]-hard and clnk-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 clnk-inapproximable on directed acyclic graphs, (4) all problems can be solved in polynomial time on trees, and (5) Directed(0,1)-Edge ((1,0)-Edge, (1,1)-Edge) Dominating Set is fixed-parameter tractable on general graphs.The first result implies that Directedr-Dominating Set on directed line graphs is NP-complete even if r=1.
Highlights
Covering and domination problems are well-studied problems in theory and in applications of graph algorithms, for example, Vertex Cover [16], Dominating Set [16] and Edge Dominating Set [24]
Vertex Cover and Edge Dominating Set on directed graphs have not been studied there are some results on directed Dominating Set [11, 7, 21, 15]
Directed (0, 1)-Edge Dominating Set is similar to Directed 1-Out Vertex Cover, surprisingly, it is NP-complete on directed acyclic graphs
Summary
Covering and domination problems are well-studied problems in theory and in applications of graph algorithms, for example, Vertex Cover [16], Dominating Set [16] and Edge Dominating Set [24]. Vertex Cover and Edge Dominating Set on directed graphs have not been studied there are some results on directed Dominating Set [11, 7, 21, 15]. Directed r-In (Out) Vertex Cover (r-In (Out) VC) is the problem that given a directed graph G = (V, E) and two positive integers k and r, determines whether there exists a vertex subset S ⊆ V of size at most k such that every edge in E is r-in (out)-covered by S. Directed (p, q)-Edge Dominating Set ((p, q)-EDS) is the problem that given a directed graph G = (V, E), one positive integer k, and two non-negative integers p, q, determines whether there exists an edge subset K ⊆ E of size at most k such that every edge is (p, q)-dominated by K. Directed (0, 1)-Edge Dominating Set is similar to Directed 1-Out Vertex Cover, surprisingly, it is NP-complete on directed acyclic graphs
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