102,371 research outputs found
On Covering Segments with Unit Intervals
We study the problem of covering a set of segments on a line with the minimum number of unit-length intervals, where an interval covers a segment if at least one of the two endpoints of the segment falls in the unit interval. We also study several variants of this problem.
We show that the restrictions of the aforementioned problems to the set of instances in which all the segments have the same length are NP-hard. This result implies several NP-hardness results in the literature for variants and generalizations of the problems under consideration.
We then study the parameterized complexity of the aforementioned problems. We provide tight results for most of them by showing that they are fixed-parameter tractable for the restrictions in which all the segments have the same length, and are W[1]-complete otherwise
Topological infinite gammoids, and a new Menger-type theorem for infinite graphs
Answering a question of Diestel, we develop a topological notion of gammoids
in infinite graphs which, unlike traditional infinite gammoids, always define a
matroid. As our main tool, we prove for any infinite graph with vertex sets
and that if every finite subset of is linked to by disjoint
paths, then the whole of can be linked to the closure of by disjoint
paths or rays in a natural topology on and its ends. This latter theorem
re-proves and strengthens the infinite Menger theorem of Aharoni and Berger for
`well-separated' sets and . It also implies the topological Menger
theorem of Diestel for locally finite graphs
Dominating sequences in grid-like and toroidal graphs
A longest sequence of distinct vertices of a graph such that each
vertex of dominates some vertex that is not dominated by its preceding
vertices, is called a Grundy dominating sequence; the length of is the
Grundy domination number of . In this paper we study the Grundy domination
number in the four standard graph products: the Cartesian, the lexicographic,
the direct, and the strong product. For each of the products we present a lower
bound for the Grundy domination number which turns out to be exact for the
lexicographic product and is conjectured to be exact for the strong product. In
most of the cases exact Grundy domination numbers are determined for products
of paths and/or cycles.Comment: 17 pages 3 figure
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