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Proximity graphs have been studied for several decades, motivated byapplications in computational geometry, geography, data mining, and many otherfields. However, the computational complexity of classic graph problems onproximity graphs mostly remained open. We now study 3-Colorability, DominatingSet, Feedback Vertex Set, Hamiltonian Cycle, and Independent Set on theproximity graph classes relative neighborhood graphs, Gabriel graphs, andrelatively closest graphs. We prove that all of the problems remain NP-hard onthese graphs, except for 3-Colorability and Hamiltonian Cycle on relativelyclosest graphs, where the former is trivial and the latter is left open.Moreover, for every NP-hard case we additionally show that no$2^{o(n^{1/4})}$-time algorithm exists unless the ETH fails, where n denotesthe number of vertices.

Most Classic Problems Remain NP-hard on Relative Neighborhood Graphs and their Relatives