Summary: This paper discusses new complexity results for several models dealing with the location of obnoxious or undesirable facilities on graphs. The focus is mainly on the continuous \(p\)-Maximin and \(p\)-Maxisum dispersion models, where the facilities can be established at the nodes or in the interiors of the edges. For the general (nonhomogeneous) case it is shown that both models are strongly \(NP\)-hard even when the underlying graph consists of a single edge. For the homogeneous \(p\)-Maximin model it is proven that even the problem of finding a \(2\over 3\)-approximation solution is \(NP\)-hard, and a polynomial heuristic which provides a \(1\over 2\)-approximation to the model is presented. Tree graphs are considered, and new algorithms with lower complexity bounds for several versions of the model are presented. For the \(p\)-Maxisum problem we show that the homogeneous case is \(NP\)- hard on general graphs. Turning to the homogeneous case on trees, a certain concavity property is identified and then utilized to improve upon the best known methods to solve this model.