In this work we start the investigation of tight complexity bounds for connectivity problems parameterized by cutwidth assuming the Strong Exponential-Time Hypothesis (SETH). Van Geffen et al. posed this question for odd cycle transversal and feedback vertex set. We answer it for these two and four further problems, namely connected vertex cover, connected domintaing set, steiner tree, and connected odd cycle transversal. For the latter two problems it sufficed to prove lower bounds that match the running time inherited from parameterization by treewidth; for the others we provide faster algorithms than relative to treewidth and prove matching lower bounds. For upper bounds we first extend the idea of Groenland et al.~[STACS~2022] to solve what we call coloring-like problem. Such problems are defined by a symmetric matrix $M$ over $\mathbb{F}_2$ indexed by a set of colors. The goal is to count the number (modulo some prime $p$) of colorings of a graph such that $M$ has a $1$-entry if indexed by the colors of the end-points of any edge. We show that this problem can be solved faster if $M$ has small rank over $\mathbb{F}_p$. We apply this result to get our upper bounds for connected vertex cover and connected dominating set. The upper bounds for odd cycle transversal and feedback vertex set use a subdivision trick to get below the bounds that matrix rank would yield.

翻译：在这项工作中,我们开始调查连接问题的严格复杂界限。 对于后两个问题, 它足以证明下限符合从树宽度参数化中继承的运行时间; 对于其他我们提供比树宽值更快的算法, 并证明匹配较低界限。 对于奇特周期跨度和反馈顶点设置, Van Geffen 等人提出了这个问题。 我们对这两个和另外四个问题作了回答, 即连接的顶点覆盖, 连接的顶点覆盖, 连接的顶点覆盖, 连接的顶点覆盖, 连接的顶点覆盖, 连接的顶点覆盖, 连接的顶点覆盖 。 这样的顶点覆盖, 将显示的顶点覆盖, 将显示的上端 美元 。 这样的顶点将显示的下端 美元 。 这样的顶点将显示的下端 美元 。 如果我们通过此端的顶点将显示的上端 水平 。