Disjointness graphs of segments in R2 are almost all hamiltonian

Let P be a set of n > 1 points in general position in R2. The edge disjointness graph D(P) of P is the graph whose vertices are all the segments with endpoints in P, two of which are adjacent in D(P) if and only if they are disjoint. In this note, we give a full characterization of all those edge disjointness graphs that are hamiltonian. More precisely, we shall show that there are exactly 9 order types of P for which D(P) is not hamiltonian. Additionally, from one of these 9 order types, we derive a counterexample to a criterion for the existence of hamiltonian cycles due to A. D. Plotnikov in 1998. (c) 2023 Elsevier B.V. All rights reserved.