  The Quadratic Travelling Salesman Problem (QTSP) is to find a least cost
Hamilton cycle in an edge-weighted graph, where costs are defined on all pairs
of edges contained in the Hamilton cycle. This is a more general version than
the commonly studied QTSP which only considers pairs of adjacent edges. We
define a restricted version of QTSP, the $k$-neighbour TSP (TSP($k$)), and give
a linear time algorithm to solve TSP($k$) on a Halin graph for $k\leq 3$. This
algorithm can be extended to solve TSP($k$) on any fully reducible class of
graphs for any fixed $k$ in polynomial time. This result generalizes
corresponding results for the standard TSP. TSP($k$) can be used to model
various machine scheduling problems as well as an optimal routing problem for
unmanned aerial vehicles (UAVs).
