10 research outputs found
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Graph Colouring with Input Restrictions
In this thesis, we research the computational complexity of the graph colouring problem and its variants including precolouring extension and list colouring for graph classes that can be characterised by forbidding one or more induced subgraphs. We investigate the structural properties of such graph classes and prove a number of new properties. We then consider to what extent these properties can be used for efficiently solving the three types of colouring problems on these graph classes. In some cases we obtain polynomial-time algorithms, whereas other cases turn out to be NP-complete.
We determine the computational complexity of k-COLOURING, k-PRECOLOURING EXTENSION and LIST k-COLOURING on -free graphs. In particular, we prove that k-COLOURING on -free graphs is NP-complete, 4-PRECOLOURING EXTENSION -free graphs is NP-complete, and LIST 4-COLOURING on -free graphs is NP-complete. In addition, we show the existence of an integer r such that k-COLOURING is NP-complete for -free graphs with girth 4. In contrast, we determine for any fixed girth a lower bound such that every -free graph with girth at least is 3-colourable. We also prove that 3-LIST COLOURING is NP-complete for complete graphs minus a matching. We present a polynomial-time algorithm for solving 4-PRECOLOURING EXTENSION on -free graphs, a polynomial-time algorithm for solving LIST 3-Colouring on -free graphs, and a polynomial-time algorithm for solving LIST 3-COLOURING on -free graphs. We prove that LIST k-COLOURING for -free graphs is also polynomial-time solvable. We obtain several new dichotomies by combining the above results with some known results
Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization
International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM
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
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum