  A $(0,1)$-matrix has the consecutive-ones property (C1P) if its columns can
be permuted to make the $1$'s in each row appear consecutively. This property
was characterised in terms of forbidden submatrices by Tucker in 1972. Several
graph classes were characterised by means of this property, including interval
graphs and strongly chordal digraphs. In this work, we define and characterise
2-nested matrices, which are $(0,1)$-matrices with a variant of the C1P and for
which there is also certain assignment of one of two colors to each block of
consecutive $1$'s in each row. The characterization of 2-nested matrices in the
present work is of key importance to characterise split graphs that are also
circle by minimal forbidden induced subgraphs.
