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.

翻译：我们研究的是多式内核的存在,在一般图形中没有多式内核的参数化问题时,我们研究的是多式内核的存在,在一般图表中没有多式内核的参数化问题时,我们研究的是,在双形图中,我们的主要结果是,双形图中,多式内核的多式内核存在多式内核的多式内核,这与标准的复杂理论假设相矛盾。减幅相当大,特别是为了将双形双形内核向下拉到4,并且可以调整为连接美元-美元内核内核的内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内,如果内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内