7 research outputs found
The general position number and the iteration time in the P3 convexity
In this paper, we investigate two graph convexity parameters: the iteration
time and the general position number. Harary and Nieminem introduced in 1981
the iteration time in the geodesic convexity, but its computational complexity
was still open. Manuel and Klav\v{z}ar introduced in 2018 the general position
number of the geodesic convexity and proved that it is NP-hard to compute. In
this paper, we extend these parameters to the P3 convexity and prove that it is
NP-hard to compute them. With this, we also prove that the iteration number is
NP-hard on the geodesic convexity even in graphs with diameter two. These
results are the last three missing NP-hardness results regarding the ten most
studied graph convexity parameters in the geodesic and P3 convexities
On the hull and interval numbers of oriented graphs
In this work, for a given oriented graph , we study its interval and hull
numbers, denoted by and , respectively, in the geodetic,
and convexities. This last one, we believe to be formally
defined and first studied in this paper, although its undirected version is
well-known in the literature. Concerning bounds, for a strongly oriented graph
, we prove that and that there is a strongly
oriented graph such that . We also determine exact
values for the hull numbers in these three convexities for tournaments, which
imply polynomial-time algorithms to compute them. These results allows us to
deduce polynomial-time algorithms to compute when the
underlying graph of is split or cobipartite. Moreover, we provide a
meta-theorem by proving that if deciding whether or
is NP-hard or W[i]-hard parameterized by , for some
, then the same holds even if the underlying graph of
is bipartite. Next, we prove that deciding whether or
is W[2]-hard parameterized by , even if the
underlying graph of is bipartite; that deciding whether or is NP-complete, even if has no directed
cycles and the underlying graph of is a chordal bipartite graph; and that
deciding whether or is W[2]-hard
parameterized by , even if the underlying graph of is split. We also
argue that the interval and hull numbers in the oriented and
convexities can be computed in polynomial time for graphs of bounded tree-width
by using Courcelle's theorem
Formulas in connection with parameters related to convexity of paths on three vertices: caterpillars and unit interval graphs
We present formulas to compute the P3 -interval number, the P3 -hull number and the percolation time for a caterpillar, in terms of certain sequences associated with it. In addition, we find a connection between the percolation time of a unit interval graph and a parameter involving the diameter of a unit interval graph related to it. Finally, we present a hereditary graph class, defined by forbidden induced subgraphs, such that its percolation time is equal to one.Fil: González, LucÃa M.. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Grippo, Luciano Norberto. Consejo Nacional de Investigaciones CientÃficas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Safe, Martin Dario. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Centro CientÃfico Tecnológico Conicet - BahÃa Blanca. Instituto de Matemática BahÃa Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática BahÃa Blanca; Argentin
Convexity in partial cubes: the hull number
We prove that the combinatorial optimization problem of determining the hull
number of a partial cube is NP-complete. This makes partial cubes the minimal
graph class for which NP-completeness of this problem is known and improves
some earlier results in the literature.
On the other hand we provide a polynomial-time algorithm to determine the
hull number of planar partial cube quadrangulations.
Instances of the hull number problem for partial cubes described include
poset dimension and hitting sets for interiors of curves in the plane.
To obtain the above results, we investigate convexity in partial cubes and
characterize these graphs in terms of their lattice of convex subgraphs,
improving a theorem of Handa. Furthermore we provide a topological
representation theorem for planar partial cubes, generalizing a result of
Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure
Complexity and algorithms related to two classes of graph problems
This thesis addresses the problems associated with conversions on graphs and editing by removing a matching. We study the f-reversible processes, which are those associated with a threshold value for each vertex, and whose dynamics depends on the number of neighbors with different state for each vertex. We set a tight upper bound for the period and transient lengths, characterize all trees that reach the maximum transient length for 2-reversible processes, and we show that determining the size of a minimum conversion set is NP-hard. We show that the AND-OR model defines a convexity on graphs. We show results of NP-completeness and efficient algorithms for certain convexity parameters for this new one, as well as approximate algorithms. We introduce the concept of generalized threshold processes, where the results are NP-completeness and efficient algorithms for both non relaxed and relaxed versions. We study the problem of deciding whether a given graph admits a removal of a matching in order to destroy all cycles. We show that this problem is NP-hard even for subcubic graphs, but admits efficient solution for several graph classes. We study the problem of deciding whether a given graph admits a removal of a matching in order to destroy all odd cycles. We show that this problem is NP-hard even for planar graphs with bounded degree, but admits efficient solution for some graph classes. We also show parameterized results.Esta tese aborda problemas associados a conversões em grafos e de edição pela remoção de um emparelhamento. Estudamos processos f-reversÃveis, que são aqueles associados a um valor de limiar para cada vértice e cuja dinâmica depende da quantidade de vizinhos com estado contrário para cada vértice. Estabelecemos um limite superior justo para o tamanho do perÃodo e transiente, caracterizamos todas as árvores que alcançam o transiente máximo em processos 2-reversÃveis e mostramos que determinar o tamanho de um conjunto conversor mÃnimo é NP-difÃcil. Mostramos que o modelo AND-OR define uma convexidade sobre grafos. Mostramos resultados de NP-completude e algoritmos eficientes para certos parâmetros de convexidade para esta nova, assim como algoritmos aproximativos. Introduzimos o conceito de processos de limiar generalizados, onde mostramos resultados de NP-completude e algoritmos eficientes para ambas as versões não relaxada e relaxada. Estudamos o problema de decidir se um dado grafo admite uma remoção de um emparelhamento de modo a remover todos os ciclos. Mostramos que este problema é NP-difÃcil mesmo para grafos subcúbicos, mas admite solução eficiente para várias classes de grafos. Estudamos o problema de decidir se um dado grafo admite uma remoção de um emparelhamento de modo a remover todos os ciclos Ãmpares. Mostramos que este problema é NP-difÃcil mesmo para grafos planares com grau limitado, mas admite solução eficiente para algumas classes de grafos. Mostramos também resultados parametrizados
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Combinatorics
Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session