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

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.

arXiv comment46 pages, 15 figures

References in corpus (5)

• Subclasses of Normal Helly Circular-Arc Graphs
• Structural characterization of some problems on circle and interval graphs
• Forbidden induced subgraph characterization of circle graphs within split graphs
• Splitting cubic circle graphs
• On nested and 2-nested graphs: two subclasses of graphs between threshold and split graphs

Cited by in corpus (1)

• Forbidden induced subgraph characterization of circle graphs within split graphs