Twin-width and Polynomial Kernels

We study the existence of polynomial kernels, for parameterized problems without a kernel on general graphs, when restricted to graphs bounded twin-width. Our main result is that k -Dominating Set twin-width at most 4 would contradict standard complexity-theoretic assumption. The reduction quite involved, especially get upper bound down 4, and can be tweaked work Connected Total (albeit with worse twin-width). -Independent problem admits same lower by much simpler argument, previously observed [ICALP ’21], which extends Dominating Set, -Path, -Induced Path, Matching, etc. On positive side, we obtain simple quadratic vertex -Vertex Cover Capacitated Interestingly applies Vapnik–Chervonenkis density 1, does not require witness sequence. also present more intricate $$O(k^{1.5})$$ Cover. Finally show deciding if graph has 1 done in time, observe optimization/decision solved time 1.