Given a digraph G, a set X ⊆ V (G) is said to be an absorbing set (resp. dominating set) if every vertex in the graph is either in X or is an in-neighbour (resp. out-neighbour) of a vertex in X. A set S ⊆ V (G) is sai...
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ISBN:
(纸本)9783959772143
Given a digraph G, a set X ⊆ V (G) is said to be an absorbing set (resp. dominating set) if every vertex in the graph is either in X or is an in-neighbour (resp. out-neighbour) of a vertex in X. A set S ⊆ V (G) is said to be an independent set if no two vertices in S are adjacent in G. A kernel (resp. solution) of G is an independent and absorbing (resp. dominating) set in G. The problem of deciding if there is a kernel (or solution) in an input digraph is known to be NP-complete. Similarly, the problems of computing a minimum cardinality kernel, absorbing set (or dominating set) and the problems of computing a maximum cardinality kernel, independent set are all known to be NP-hard for general digraphs. We explore the algorithmic complexity of these problems in the well known class of interval digraphs. A digraph G is an interval digraph if a pair of intervals (Su, Tu) can be assigned to each vertex u of G such that (u, v) ∈ E(G) if and only if Su ∩ Tv ≠ ∅. Many different subclasses of interval digraphs have been defined and studied in the literature by restricting the kinds of pairs of intervals that can be assigned to the vertices. We observe that several of these classes, like interval catch digraphs, interval nest digraphs, adjusted interval digraphs and chronological interval digraphs, are subclasses of the more general class of reflexive interval digraphs – which arise when we require that the two intervals assigned to a vertex have to intersect. We see as our main contribution the identification of the class of reflexive interval digraphs as an important class of digraphs. We show that all the problems mentioned above are efficiently solvable, in most of the cases even linear-time solvable, in the class of reflexive interval digraphs, but are APX-hard on even the very restricted class of interval digraphs called point-point digraphs, where the two intervals assigned to each vertex are required to be degenerate, i.e. they consist of a single point each. The re
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