On the Connectivity of the Flip Graph of Plane Spanning Paths

Flip graphs of non-crossing configurations in the plane are widely studied objects, e.g., flip graph of triangulations, spanning trees, Hamiltonian cycles, and perfect matchings. Typically, it is an easy exercise to prove connectivity of a flip graph. In stark contrast, the connectivity of the flip graph of plane spanning paths on point sets in general position has been an open problem for more than 16 years. In order to provide new insights, we investigate certain induced subgraphs. Firstly, we provide tight bounds on the diameter and the radius of the flip graph of spanning paths on points in convex position with one fixed endpoint. Secondly, we show that so-called suffix-independent paths induce a connected subgraph. Consequently, to answer the open problem affirmatively, it suffices to show that each path can be flipped to some suffix-independent path. Lastly, we investigate paths where one endpoint is fixed and provide tools to flip to suffix-independent paths. We show that these tools are strong enough to show connectivity of the flip graph of plane spanning paths on point sets with at most two convex layers.