We consider the problem of locating a single facility on a vertex in a given graph based on agents’ preferences, where the domain of the preferences is either single-peaked or single-dipped, depending on whether they want to access the facility (a public good) or be far from it (a public bad). Our main interest is the existence of deterministic social choice functions that are Pareto efficient and false-name-proof, i.e., resistant to fake votes. We show that regardless of whether preferences are single-peaked or single-dipped, such a social choice function exists (i) for any tree graph, and (ii) for a cycle graph if and only if its length is less than six. We also show that when the preferences are single-peaked, such a social choice function exists for any ladder (i.e., 2 × m grid) graph, and does not exist for any larger (hyper)grid.