45,848 research outputs found
On the recognition and characterization of M-partitionable proper interval graphs
For a symmetric {0, 1, ⋆ }-matrix M of size m, a graph G is said to be M-partitionable, if its vertices can be partitioned into sets V1, V2, . . . , Vm, such that two parts Vi, Vj are completely adjacent if Mi,j = 1, and completely non-adjacent if Mi,j = 0 (Vi is considered completely adjacent to itself if it induces a clique, and completely non-adjacent if it induces an independent set). The complexity problem (or the recognition problem) for a matrix M asks whether the M-partition problem is polynomial-time solvable or NP-complete. The characterization problem for a matrix M asks if all M-partitionable graphs can be characterized by the absence of a finite set of forbidden induced subgraphs. These forbidden induced subgraphs are called obstructions to M. In the literature, many results were obtained by restricting the input graphs. In this thesis, we survey these results when the questions are restricted to the class of perfect graphs. We then study the recognition problem and the characterization problem when the inputs are restricted to proper interval graphs. The recognition problem can be solved by an existing algorithm, but we simplify its proof of correctness. As our main result, we prove that all the matrices of size 3 and size 4 with constant diagonal, have finitely many minimal proper interval obstructions. We also obtain partial results about matrices of arbitrary size if they have a zero diagonal
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 -matrix has the
consecutive-ones property (C1P) for the rows if there is a permutation of its
columns such that the 's in each row appear consecutively. In this thesis,
we develop characterizations by forbidden subconfigurations of -matrices
with the C1P for which the rows are -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 , a
-completion of a graph is a graph such
that belongs to . A -completion of is minimal if does not belong to for every proper subset of . A
-completion of is minimum if for every -completion of , the cardinal of is less than or equal to the cardinal
of . 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
Subclasses of Normal Helly Circular-Arc Graphs
A Helly circular-arc model M = (C,A) is a circle C together with a Helly
family \A of arcs of C. If no arc is contained in any other, then M is a proper
Helly circular-arc model, if every arc has the same length, then M is a unit
Helly circular-arc model, and if there are no two arcs covering the circle,
then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit
Helly, normal Helly) circular-arc graph is the intersection graph of the arcs
of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model.
In this article we study these subclasses of Helly circular-arc graphs. We show
natural generalizations of several properties of (proper) interval graphs that
hold for some of these Helly circular-arc subclasses. Next, we describe
characterizations for the subclasses of Helly circular-arc graphs, including
forbidden induced subgraphs characterizations. These characterizations lead to
efficient algorithms for recognizing graphs within these classes. Finally, we
show how do these classes of graphs relate with straight and round digraphs.Comment: 39 pages, 13 figures. A previous version of the paper (entitled
Proper Helly Circular-Arc Graphs) appeared at WG'0
Perfect Elimination Orderings for Symmetric Matrices
We introduce a new class of structured symmetric matrices by extending the
notion of perfect elimination ordering from graphs to weighted graphs or
matrices. This offers a common framework capturing common vertex elimination
orderings of monotone families of chordal graphs, Robinsonian matrices and
ultrametrics. We give a structural characterization for matrices that admit
perfect elimination orderings in terms of forbidden substructures generalizing
chordless cycles in graphs.Comment: 16 pages, 3 figure
A polynomial algorithm for the k-cluster problem on interval graphs
This paper deals with the problem of finding, for a given graph and a given
natural number k, a subgraph of k nodes with a maximum number of edges. This
problem is known as the k-cluster problem and it is NP-hard on general graphs
as well as on chordal graphs. In this paper, it is shown that the k-cluster
problem is solvable in polynomial time on interval graphs. In particular, we
present two polynomial time algorithms for the class of proper interval graphs
and the class of general interval graphs, respectively. Both algorithms are
based on a matrix representation for interval graphs. In contrast to
representations used in most of the previous work, this matrix representation
does not make use of the maximal cliques in the investigated graph.Comment: 12 pages, 5 figure
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