On nested and 2-nested graphs: two subclasses of graphs between threshold and split graphs

Abstract: A $(0,1)$-matrix has the Consecutive Ones Property (C1P) for the rows if there is a permutation of its columns such that the ones in each row appear consecutively. We say a $(0, 1)$-matrix is nested if it has the consecutive ones property for the rows (C1P) and every two rows are either disjoint or nested. We say a $(0, 1)$-matrix is 2-nested if it has the C1P and admits a partition of its rows into two sets such that the submatrix induced by each of these sets is nested. We say a split graph $G$ with split partition $(K, S)$ is nested (resp.\ 2-nested) if the matrix $A(S, K)$ which indicates the adjacency between vertices in $S$ and $K$ is nested (resp.\ 2-nested). In this work, we characterize nested and 2-nested matrices by minimal forbidden submatrices. This characterization leads to a minimal forbidden induced subgraph characterization for these classes of graphs, which are a superclass of threshold graphs and a subclass of split and circle graphs.