Before revision: 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 ).
After revision: We unify and extend previous kernelization techniques in sparse classes [ 6,17 ] 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 ).