  In 1969, Alan Tucker characterized proper circular-arc graphs as those graphs
whose augmented adjacency matrices have the circularly compatible ones
property. Moreover, he also found a polynomial-time algorithm for deciding
whether any given augmented adjacency matrix has the circularly compatible ones
property. These results allowed him to devise the first polynomial-time
recognition algorithm for proper circular-arc graphs. However, as Tucker
himself remarks, he did not solve the problems of finding a structure theorem
and an efficient recognition algorithm for the circularly compatible ones
property in arbitrary matrices (i.e., not restricted to augmented adjacency
matrices only). In this work, we solve these problems. More precisely, we give
a minimal forbidden submatrix characterization for the circularly compatible
ones property in arbitrary matrices and a linear-time recognition algorithm for
the same property. We derive these results from analogous ones for the related
$D$-circular property. Interestingly, these results lead to a minimal forbidden
induced subgraph characterization and a linear-time recognition algorithm for
proper circular-arc bigraphs, solving a problem first posed by Basu, Das,
Ghosh, and Sen [J. Graph Theory, 73(4):361--376, 2013]. Our findings generalize
some known results about $D$-interval hypergraphs and proper interval bigraphs.
