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 $τ^2+τ$ vertices for GTSP, where $τ$ is the vertex cover number of the input graph. Any $α$-approximate solution for the problem kernel also gives an $α$-approximate solution for the original instance, for any $α\geq1$.