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 each vertex in an undirected edge-weighted and not necessarily complete graph. We present a problem kernel with ${\tau }^2+\tau$ vertices for GTSP, where $\tau$ is the vertex cover number of the input graph. Any $\alpha$-approximate solution for the problem kernel also gives an $\alpha$-approximate solution for the original instance, for any $\alpha \ge 1$.