6 research outputs found
Linear kernels for outbranching problems in sparse digraphs
In the -Leaf Out-Branching and -Internal Out-Branching problems we are
given a directed graph with a designated root and a nonnegative integer
. The question is to determine the existence of an outbranching rooted at
that has at least leaves, or at least internal vertices,
respectively. Both these problems were intensively studied from the points of
view of parameterized complexity and kernelization, and in particular for both
of them kernels with vertices are known on general graphs. In this
work we show that -Leaf Out-Branching admits a kernel with vertices
on -minor-free graphs, for any fixed family of graphs
, whereas -Internal Out-Branching admits a kernel with
vertices on any graph class of bounded expansion.Comment: Extended abstract accepted for IPEC'15, 27 page
Kernelization and Sparseness: the case of Dominating Set
We prove that for every positive integer and for every graph class
of bounded expansion, the -Dominating Set problem admits a
linear kernel on graphs from . Moreover, when is only
assumed to be nowhere dense, then we give an almost linear kernel on for the classic Dominating Set problem, i.e., for the case . These
results generalize a line of previous research on finding linear kernels for
Dominating Set and -Dominating Set. However, the approach taken in this
work, which is based on the theory of sparse graphs, is radically different and
conceptually much simpler than the previous approaches.
We complement our findings by showing that for the closely related Connected
Dominating Set problem, the existence of such kernelization algorithms is
unlikely, even though the problem is known to admit a linear kernel on
-topological-minor-free graphs. Also, we prove that for any somewhere dense
class , there is some for which -Dominating Set is
W[]-hard on . Thus, our results fall short of proving a sharp
dichotomy for the parameterized complexity of -Dominating Set on
subgraph-monotone graph classes: we conjecture that the border of tractability
lies exactly between nowhere dense and somewhere dense graph classes.Comment: v2: new author, added results for r-Dominating Sets in bounded
expansion graph
Linear Kernels for Outbranching Problems in Sparse Digraphs
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