4 research outputs found
On the Kernel and Related Problems in Interval Digraphs
Given a digraph , a set is said to be absorbing set
(resp. dominating set) if every vertex in the graph is either in or is an
in-neighbour (resp. out-neighbour) of a vertex in . A set
is said to be an independent set if no two vertices in are adjacent in .
A kernel (resp. solution) of is an independent and absorbing (resp.
dominating) set in . We explore the algorithmic complexity of these problems
in the well known class of interval digraphs. A digraph is an interval
digraph if a pair of intervals can be assigned to each vertex
of such that if and only if .
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 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 results we obtain improve and
generalize several existing algorithms and structural results for subclasses of
reflexive interval digraphs.Comment: 26 pages, 3 figure
Interval Digraphs and Bounded Bitolerance Digraphs
Interval digraphs and bounded bitolerance digraphs are two
different directed graph analogues to the well-known interval graphs.
Both are contained in the class of digraphs of Ferrers Dimension
at most two. In this paper we explore the relationship between
these two important classes of digraphs