  A symmetric matrix is Robinsonian if its rows and columns can be
simultaneously reordered in such a way that entries are monotone nondecreasing
in rows and columns when moving toward the diagonal. The adjacency matrix of a
graph is Robinsonian precisely when the graph is a unit interval graph, so that
Robinsonian matrices form a matrix analogue of the class of unit interval
graphs. Here we provide a structural characterization for Robinsonian matrices
in terms of forbidden substructures, extending the notion of asteroidal triples
to weighted graphs. This implies the known characterization of unit interval
graphs and leads to an efficient algorithm for certifying that a matrix is not
Robinsonian.
