Row and column orthogonal (0,1)-matrices

In this note we investigate pairs of graphs that can be realized as the row-orthogonal and column-orthogonal graphs of a square (0,1)-matrix, and graphs that can be realized as the row- and column-orthogonal graph of a symmetric (0,1)-matrix. We provide a general construction that shows, in particular, that trees, and all bipartite graphs with a vertex of degree 1 or of diameter at least 9 have such a symmetric realization.