A quadratic-order problem kernel for the traveling salesman problem parameterized by the vertex cover number

The NP-hard graphical traveling salesman problem (GTSP) is to find a closed walk of total minimum weight that visits (at least once) each vertex in an undirected edge-weighted and not necessarily complete graph. Recently, Bla\v{z}ej et al. [ESA 2022] showed a problem kernel with $O(\tau^3)$ vertices for GTSP, where $\tau$ is the vertex cover number of the input graph. We present a problem kernel with only $O(\tau^2)$ vertices.