Twin-width and Polynomial Kernels

We study the existence of polynomial kernels, for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. Our main result is that a polynomial kernel for k- DOMINATING SET on graphs of twin-width at most 4 would contradict a standard complexity-theoretic assumption. The reduction is quite involved, especially to get the twin-width upper bound down to 4, and can be tweaked to work for CONNECTED k- DOMINATING SET and TOTAL k- DOMINATING SET (albeit with a worse upper bound on the twin-width). The k-INDEPENDENT SET problem admits the same lower bound by a much simpler argument, previously observed [ICALP '21], which extends to k- INDEPENDENT DOMINATING SET, k- PATH, k- INDUCED PATH, k- INDUCED MATCHING, etc. On the positive side, we obtain a simple quadratic vertex kernel for CONNECTED k- VERTEX COVER and CAPACITATED k- VERTEX COVER on graphs of bounded twin-width. Interestingly the kernel applies to graphs of Vapnik-Chervonenkis density 1, and does not require a witness sequence. We also present a more intricate O(k(1.5)) vertex kernel for CONNECTED k- VERTEX COVER. Finally we show that deciding if a graph has twin-width at most 1 can be done in polynomial time, and observe that most optimization/decision graph problems can be solved in polynomial time on graphs of twin-width at most 1.