# 2-nested matrices: towards understanding the structure of circle graphs

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.

## Authors

• 1 publication
• 4 publications
• 6 publications
• ### On nested and 2-nested graphs: two subclasses of graphs between threshold and split graphs

A (0,1)-matrix has the Consecutive Ones Property (C1P) for the rows if t...
06/27/2019 ∙ by Nina Pardal, et al. ∙ 0

• ### Structural characterization of some problems on circle and interval graphs

A graph is circle if there is a family of chords in a circle such that t...
05/29/2020 ∙ by Nina Pardal, et al. ∙ 0

• ### Fast Algorithms for Indices of Nested Split Graphs Approximating Real Complex Networks

We present a method based on simulated annealing to obtain a nested spli...
03/01/2018 ∙ by Irene Sciriha, et al. ∙ 0

• ### Strongly chordal digraphs and Γ-free matrices

We define strongly chordal digraphs, which generalize strongly chordal g...
09/09/2019 ∙ by Pavol Hell, et al. ∙ 0

• ### Minimal obstructions for a matrix partition problem in chordal graphs

If M is an m × m matrix over { 0, 1, ∗}, an M-partition of a graph G is ...
04/02/2020 ∙ by Juan Carlos García-Altamirano, et al. ∙ 0

• ### Circularly compatible ones, D-circularity, and proper circular-arc bigraphs

In 1969, Alan Tucker characterized proper circular-arc graphs as those g...
06/02/2019 ∙ by Martín D. Safe, et al. ∙ 0