Abstract: This paper proposes a novel message passing neural (MPN) architecture Conv-MPN, which reconstructs an outdoor building as a planar graph from a single RGB image. Conv-MPN is specifically designed for cases where nodes of a graph have explicit spatial embedding. In our problem, nodes correspond to building edges in an image. Conv-MPN is different from MPN in that 1) the feature associated with a node is represented as a feature volume instead of a 1D vector. and 2) convolutions encode messages instead of fully connected layers. Conv-MPN learns to select a true subset of nodes (i.e., building edges) to reconstruct a building planar graph. Our qualitative and quantitative evaluations over 2,000 buildings show that Conv-MPN makes significant improvements over the existing fully neural solutions. We believe that the paper has a potential to open a new line of graph neural network research for structured geometry reconstruction.