# 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 two vertices are adjacent if the corresponding chords cross each other. There are diverse characterizations of circle graphs, many of them using the notions of local complementation or split decomposition. However, there are no known structural characterization by minimal forbidden induced subgraphs for circle graphs. In this thesis, we give a characterization by forbidden induced subgraphs of circle graphs within split graphs. A (0,1)-matrix has the consecutive-ones property (C1P) for the rows if there is a permutation of its columns such that the 1's in each row appear consecutively. In this thesis, we develop characterizations by forbidden subconfigurations of (0,1)-matrices with the C1P for which the rows are 2-colorable under a certain adjacency relationship, and we characterize structurally some auxiliary circle graph subclasses that arise from these special matrices. Given a graph class Π, a Π-completion of a graph G = (V,E) is a graph H = (V, E ∪ F) such that H belongs to Π. A Π-completion H of G is minimal if H'= (V, E ∪ F') does not belong to Π for every proper subset F' of F. A Π-completion H of G is minimum if for every Π-completion H' = (V, E ∪ F') of G, the cardinal of F is less than or equal to the cardinal of F'. In this thesis, we study the problem of completing minimally to obtain a proper interval graph when the input is an interval graph. We find necessary conditions that characterize a minimal completion in this particular case, and we leave some conjectures for the future.

## Authors

• 4 publications
• ### Forbidden induced subgraph characterization of circle graphs within split graphs

A graph is circle if its vertices are in correspondence with a family of...
07/01/2020 ∙ by Flavia Bonomo-Braberman, et al. ∙ 0

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

A (0,1)-matrix has the consecutive-ones property (C1P) if its columns ca...
03/05/2021 ∙ by Guillermo Durán, et al. ∙ 0

• ### 𝒮-adic characterization of minimal ternary dendric subshifts

Dendric subshifts are defined by combinatorial restrictions of the exten...
02/19/2021 ∙ by France Gheeraert, 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

• ### Daisy cubes: a characterization and a generalization

Daisy cubes are a recently introduced class of isometric subgraphs of hy...
05/17/2019 ∙ by Andrej Taranenko, et al. ∙ 0

• ### 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