FundaÃÃo Cearense de Apoio ao Desenvolvimento Cientifico e TecnolÃgico / In this dissertation we present complexity results related to the hull number and the convexity number for P3 convexity. We show that the hull number and the convexity number are NP-hard even for bipartite graphs. Inspired by our research in convexity based on paths, we introduce a new convexity, where we defined as convexity of induced paths of order three or P∗ 3 . We show a relation between the geodetic convexity and the P∗ 3 convexity when the graph is a join of a Km with a non-complete graph. We did research in geometric convexity and from that we characterized graph classes under some convexities such as the star florest in P3 convexity, chordal cographs in P∗ 3 convexity, and the florests in TP convexity. We also demonstrated convexities that are geometric only in specific graph classes such as cographs in P4+-free convexity, F free graphs in F-free convexity and others. Finally, we demonstrated some results of geodesic convexity and P∗ 3 in graphs with few P4âs. / In this dissertation we present complexity results related to the hull number and the convexity number for P3 convexity. We show that the hull number and the convexity number are NP-hard even for bipartite graphs. Inspired by our research in convexity based on paths, we introduce a new convexity, where we defined as convexity of induced paths of order three or P∗ 3 . We show a relation between the geodetic convexity and the P∗ 3 convexity when the graph is a join of a Km with a non-complete graph. We did research in geometric convexity and from that we characterized graph classes under some convexities such as the star florest in P3 convexity, chordal cographs in P∗ 3 convexity, and the florests in TP convexity. We also demonstrated convexities that are geometric only in specific graph classes such as cographs in P4+-free convexity, F free graphs in F-free convexity and others. Finally, we demonstrated some results of geodesic convexity and P∗ 3 in graphs with few P4âs.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / In this work, we study some parameters of monophonic convexity in some classes of graphs and we present our results about this subject. We prove that decide if the $m$-interval number is at most 2 and decide if the $m$-percolation time is at most 1 are NP-complete problems even on bipartite graphs. We also prove that the $m$-convexity number is as hard to approximate as the maximum clique problem, which is, $O(n^{1-varepsilon})$-unapproachable in polynomial-time, unless P=NP, for each $varepsilon>0$. Finally, we obtain polynomial time algorithms to compute the $m$-convexity number on hereditary graph classes such that the computation of the clique number is polynomial-time solvable (e.g. perfect graphs and planar graphs). / Neste trabalho, estudamos alguns parÃmetros para a convexidade monofÃnica em algumas classes de grafos e apresentamos nossos resultados acerca do assunto. Provamos que decidir se o nÃmero de $m$-intervalo à no mÃximo 2 e decidir se o tempo de $m$-percolaÃÃo à no mÃximo 1 sÃo problemas NP-completos mesmo em grafos bipartidos. TambÃm provamos que o nÃmero de $m$-convexidade à tÃo difÃcil de aproximar quanto o problema da Clique MÃxima, que Ã, $O(n^{1-varepsilon})$-inaproximÃvel em tempo polinomial, a menos que P=NP, para cada $varepsilon>0$. Finalmente, apresentamos um algoritmo de tempo polinomial para determinar o nÃmero de $m$-convexidade em classes hereditÃrias de grafos onde a computaÃÃo do tamanho da clique mÃxima à em tempo polinomial (como grafos perfeitos e grafos planares).