Jones' conjecture for Halin graphs and a bit more

We prove Jones' famous conjecture for Halin graphs and a somewhat more general class of graphs too. A based planar graph is a planar one that has a face adjacent to every other face. We confirm Jones' conjecture for based planar graphs. Namely, if a based planar graph does not contain k+1 vertex-disjoint cycles, then it suffices to delete 2k vertices to make it acyclic.