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The Quadratic Travelling Salesman Problem (QTSP) is to find a least costHamilton cycle in an edge-weighted graph, where costs are defined on all pairsof edges contained in the Hamilton cycle. This is a more general version thanthe commonly studied QTSP which only considers pairs of adjacent edges. Wedefine a restricted version of QTSP, the $k$-neighbour TSP (TSP($k$)), and givea linear time algorithm to solve TSP($k$) on a Halin graph for $k\leq 3$. Thisalgorithm can be extended to solve TSP($k$) on any fully reducible class ofgraphs for any fixed $k$ in polynomial time. This result generalizescorresponding results for the standard TSP. TSP($k$) can be used to modelvarious machine scheduling problems as well as an optimal routing problem forunmanned aerial vehicles (UAVs).

A Linear Time Algorithm for the $3$-neighbour Traveling Salesman Problem on Halin graphs and extensions