Sobre grafos com r tamanhos diferentes de conjuntos independentes maximais e algumas extensões

• Autor(a) principal: Cappelle, Márcia Rodrigues
• Data de Publicação: 2014
• Tipo de documento: Tese
• Idioma: por
• Título da fonte: Repositório Institucional da UFG
• dARK ID: ark:/38995/0013000001x11
• Texto Completo: http://repositorio.bc.ufg.br/tede/handle/tede/4475
• topic: Teoria dos grafos; Conjuntos independentes; Grafos bem-cobertos; Produtos complementares; Graph theory; Independent sets; Well-covered graphs; Complementary products; CIENCIAS EXATAS E DA TERRA::CIENCIA DA COMPUTACAO

In this thesis, we present some results concerning about the sizes of maximal independent sets in graphs. We prove that for integers r and D with r 2 and D 3, there are only finitely many connected graphs of minimum degree at least 2, maximum degree at most D, and girth at least 7 that have maximal independent sets of at most r different sizes. Furthermore, we prove several results restricting the degrees of such graphs. These contributions generalize known results on well-covered graphs. We study the structure and recognition of the well-covered graphs G with order n(G) without an isolated vertex that have independence number n(G)k 2 for some non-negative integer k. For k = 1, we give a complete structural description of these graphs, and for a general but fixed k, we describe a polynomial time recognition algorithm. We consider graphs G without an isolated vertex for which the independence number a(G) and the independent domination number i(G) satisfy a(G) i(G) k for some non-negative integer k. We obtain a upper bound on the independence number in these graphs. We present a polynomial algorithm to recognize some complementary products, which includes all complementary prisms. Also, we present results on well-covered complementary prisms. We show that if G is not well-covered and its complementary prism is well-covered, then G has only two consecutive sizes of maximal independent sets. We present an upper bound for the quantity of sizes of maximal independent sets in complementary prisms and other wellcovered concerning results. We present a lower bound for the quantity of different sizes of maximal independent sets in Cartesian products of paths and cycles.