Atribuição de papéis em alguns produtos de grafos

• Autor(a) principal: Mesquita, Fernanda Neiva
• Data de Publicação: 2022
• Tipo de documento: Tese
• Idioma: por
• Título da fonte: Repositório Institucional da UFG
• Texto Completo: http://repositorio.bc.ufg.br/tede/handle/tede/12185
• topic: Atribuição de papéis; Produto cartesiano; Prisma complementar; Redes sociais; Role assignment; Cartesian product; Complementary prism; Social networks; CIENCIAS EXATAS E DA TERRA::CIENCIA DA COMPUTACAO::TEORIA DA COMPUTACAO

During the pandemic, due to the new coronavirus (COVID-19), the use of social networks was enhanced by social distancing and the need to stay connected, generating a gigantic volume of data. In order to extract information, graphs constitute a powerful modeling tool in which the vertices represent individuals and the edges represent relationships between them. In 1991, Everett and Borgatti formalized the concept of role assignment under the name role coloring. Thus, a r-role assignment of a simple graph G is an assignment of r distinct roles to the vertices of G, such that two vertices with the same role have the same set of roles in the related vertices. Furthermore, a specific r-role assignment defines a role graph, in which the vertices are the distinct r roles, and there is an edge between two roles whenever there are two related vertices in the graph G that correspond to these roles. Research on role assignment and operations on graphs is scarce. We showed a dichotomy for the r-role assignment problem for the Cartesian product. While the Cartesian product of two graphs always admits a 2-role assignment, the problem remains NP-complete for any fixed r ≥ 3. The complementary prism arises from the complementary product, introduced by Haynes, Henning and Van Der Merwe in 2019, which is a generalization of the Cartesian product. Complementary prisms admits a 2-role assignment, with the exception of the complementary prism of a path with three vertices. We verified that the complementary prisms admits a 3-role assignment, with the exception of the complementary prism of some not connected bipartite graphs. Next, we showed that the related problem can be solved in linear time. Finally, we conjecture that, for r ≥ 3 the problem of (r+1)-role assignment to complementary prisms is NP-complete. In this sense, we consider the role graph K'_{1,r} which is the bipartite graph K_{1,r} with a loop at the vertex of degree r and we highlight that the problem of deciding whether a prism complement has a (r+1)-role assignment, when the role graph is K'_{1,r}, it is NP-complete.