We unify and extend previous kernelization techniques in sparse classes [ 5,16 ] by defining a structure we call water lilies and show how it can be used in bounded expansion classes to construct linear bikernels for (r, c) -DOMINATING SET , (r, c) -SCATTERED SET, TOTAL r-DOMINATION , r-Roman Domination, and a problem we call (r, [{\lambda}, {\mu}]) -DOMINATION (implying a bikernel for r-PERFECT CODE ). At the cost of slightly changing the output graph class these bikernels can be turned into into kernels. Finally, we demonstrate how these constructions can be combined to create 'multikernels', meaning graphs who represent kernels for multiple problems at once. Concretely, we show that r-DOMINATING SET, TOTAL r-DOMINATION, and r-ROMAN DOMINATION admit a multikernel; as well as r-DOMINATING SET and 2r-INDEPENDENT SET for multiple values of r at once.
We unify and extend previous kernelization techniques in sparse classes [ 6,17 ] by defining water lilies and show how they can be used in bounded expansion classes to construct linear bikernels for (r, c) -Dominating Set , (r, c) -Scattered Set, Total r-Domination , r-Roman Domination, and a problem we call (r, [{\lambda}, {\mu}]) -Domination (implying a bikernel for r-Perfect Code ). At the cost of slightly changing the output graph class our bikernels can be turned into kernels. We further demonstrate how these constructions can be combined to create 'multikernels', meaning graphs that represent kernels for multiple problems at once. Concretely, we show that r-Dominating Set, Total r-Domination, and r-Roman Domination admit a multikernel; as well as r-Dominating Set and 2r-Independent Set for multiple values of r at once.
R
5,16
6,17
D
a structure we call
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it
they
R
-DOMINATING SET
-Dominating Set
R
-SCATTERED SET, TOTAL r-DOMINATION
-Scattered Set, Total r-Domination
R
-DOMINATION
-Domination
R
r-PERFECT CODE
r-Perfect Code
R
these
our
R
into kernels. Finally, we
kernels. We further
R
who
that
R
r-DOMINATING SET, TOTAL r-DOMINATION, and r-ROMAN DOMINATION
r-Dominating Set, Total r-Domination, and r-Roman Domination
R
r-DOMINATING SET and 2r-INDEPENDENT SET
r-Dominating Set and 2r-Independent Set