Persistency in the Traveling Salesman Problem on Halin graphs

For the Traveling Salesman Problem (TSP) on Halin graphs with three types of cost functions: sum, bottleneck and balanced and with arbitrary real edge costs we compute in polynomial time the persistency partition $E_{All}$, $E_{Some}$, $E_{None}$ of the edge set E, where:
$E_{All}$ = {e ∈ E, e belongs to all optimum solutions},
$E_{None}$ = {e ∈ E, e does not belong to any optimum solution} and
$E_{Some}$ = {e ∈ E, e belongs to some but not to all optimum solutions}.