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

TL;DR

This paper introduces and characterizes 2-nested matrices, a new class of matrices related to the consecutive-ones property, to better understand the structure of circle graphs and their associated split graphs.

Contribution

It defines and characterizes 2-nested matrices, extending the C1P property with a coloring scheme, to aid in characterizing circle graphs and split graphs.

Findings

Characterization of 2-nested matrices

Connection to circle graph structure

Identification of minimal forbidden subgraphs

Abstract

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.