Eulerian and Hamiltonian Graphs · Hard problem

The wheel graph $W_6$ has a hub $H$ joined to all $6$ vertices of a $6$-cycle rim, plus the $6$ rim edges ($12$ edges total, every vertex odd degree). What is the minimum number of EDGES that must be deleted (keeping the graph spanning all $7$ vertices and connected) so the result has an Euler TRAIL (allowing up to $2$ odd vertices, not necessarily $0$)?