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 $H = (V, E \cup F)$ such that $H$ belongs to $Π$. A $Π$-completion $H$ of $G$ is minimal if $H'= (V, E \cup F')$ 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 $H' = (V, E \cup F')$ 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.

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