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
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
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-reversĂveis, que sĂŁo aqueles associados a um valor de limiar para cada vĂ©rtice e cuja dinâmica depende da quantidade de vizinhos com estado contrário para cada vĂ©rtice. Estabelecemos um limite superior justo para o tamanho do perĂodo e transiente, caracterizamos todas as árvores que alcançam o transiente máximo em processos 2-reversĂveis e mostramos que determinar o tamanho de um conjunto conversor mĂnimo Ă© NP-difĂcil. 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 nĂŁo 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 Ă© NP-difĂcil mesmo para grafos subcĂşbicos, 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 Ămpares. Mostramos que este problema Ă© NP-difĂcil mesmo para grafos planares com grau limitado, mas admite solução eficiente para algumas classes de grafos. Mostramos tambĂ©m resultados parametrizados