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 Î , a
Î -completion of a graph G=(V,E) is a graph H=(V,EâȘF) such
that H belongs to Î . A Î -completion H of G is minimal if HâČ=(V,EâȘFâČ) does not belong to Î for every proper subset FâČ of F. A
Î -completion H of G is minimum if for every Î -completion HâČ=(V,EâȘFâČ) of G, the cardinal of F is less than or equal to the cardinal
of FâČ. 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.Comment: PhD Thesis, joint supervision Universidad de Buenos
Aires-Universit\'e Paris-Nord. Dissertation took place on March 30th 202