Multidimensional matrices uniquely recovered by their lines

We provide a method to determine if a q-ary multidimensional matrix is lonesum or not by using properties of line sums of lonesum multidimensional matrices. In particular, we establish a graphic method that uses edge-colored graphs to determine if a binary multidimensional matrix is lonesum or not. We also provide two methods to determine if a q-ary multidimensional matrix is lonestructure or not. The first one uses properties of line structures of lonestructure multidimensional matrices and the second one uses edge-colored directed multigraphs.