We introduce and study two natural generalizations of the Connected VertexCover (VC) problem: $p$-Edge-Connected $p$-Vertex-Connected VC problem (where $p \geq 2$ is a fixed integer). Like VC, both new problems are FPT, but do not admit polynomial kernel unless $NP \subseteq coNP/poly$, which highly unlikely. prove however that time efficient sized approximate kernelization schemes. obtain an $...

We relate commutative algebras in braided tensor categories to braid-reversed equivalences, motivated by vertex algebra representation theory. First, for $\mathcal{C}$ a category, we give detailed construction of the canonical $\mathcal{C}\boxtimes\mathcal{C}^\text{rev}$: if is semisimple but not necessarily finite or rigid, then $\bigoplus_{X\in\text{Irr}(\mathcal{C})}X'\boxtimes X$ algebra, w...