We present a novel deadlock analysis for concurrent objects based on the results inferred by a points-to analysis and a may-happen-in-parallel (MHP) analysis. Similarly to other analysis, we build a dependency graph such that the absence of cycles in the graph ensures deadlock freeness. An MHP analysis provides an over-approximation of the pairs of program points that may be running in parallel. The crux of the method is that the analysis integrates the MHP information within the dependency graph in order to discard unfeasible cycles that otherwise would lead to false positives. We argue that our analysis is more precise and/or efficient than previous proposals for deadlock analysis of concurrent objects. As regards accuracy, we are able to handle cases that other analyses have pointed out as challenges. As regards efficiency, the complexity of our deadlock analysis is polynomial.