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Eta-perfect graphs : thesis submitted in partial fulfillment of the requirements for the degree of Master of Science
Aleš, Janez, 1966-
Graph parameters ▫$\eta(G)$▫ and ▫$\gamma(G)$▫ are studied in this thesis. ▫$\eta(G)$▫ denite the maximum number of disjoint closed neighbourhoods in a graph ▫$G$▫ and ▫$\gamma(G)$▫ denote the ...cardinality of a minimum dominating set of a graph ▫$G$▫. We prove Gaddum-Nordhaus type results for the parameter ▫$\eta(G)$▫ and some other inequalities concerning ▫$\eta(G)$▫ and ▫$\gamma(G)$▫. Perfect graphs were defined by C. Berge. He introduced the terms ▫$\alpha$▫-perfect graphs and ▫$\omega$▫-perfect graphs. A proof of the Weak Perfect Graph Conjecture is given. A class of ▫$\eta$▫-perfect graphs is introduced in a way similar to the definition of ▫$\alpha$▫-perfect graphs. A graph ▫$G$▫ is ▫$\eta$▫-perfect for all ▫$U \subseteq V(G)$▫, ▫$\eta(G_U) = \gamma(G_U)$▫. Concepts of perfect and ▫$\eta$▫-perfect graphs are related, and some classes of ▫$\eta$▫-perfect graphs which are also perfect are studied. Strongly chordal graphs, interval graphs, trees, line graphs of trees, and total graphs of trees are proven to be ▫$\eta$▫-perfect. Parameters ▫$\eta(G)$▫ and ▫$\gamma(G)$▫ are computed gor paths, cycles, cliques and trampolines. The paths ▫$P_n$▫, the cycles ▫$C_{3n}$▫, and the trampolines ▫$T_{2n}$▫ are proven to be ▫$\eta$▫-perfect for any integer ▫$n \ge 1$▫. Computation of parameters ▫$\eta(G)$▫ and ▫$\gamma(G)$▫ is also studied in terms of linear programming. A class of ▫$\eta$▫-critical graphs is defined. A graph ▫$G$▫ is ▫$\eta$▫-critical if ▫$G$▫ is not ▫$\eta$▫-perfect and ▫$G_{V(G)-v}$▫ is ▫$\eta$▫-perfect for every ▫$v \in V(G)$▫. The cycles ▫$C_{3n+1}$▫, ▫$C_{3n+2}$▫ and the trampolines ▫$T_{2n+1}$▫ are proven to be ▫$\eta$▫-critical for any integer ▫$n \ge 1$▫.
Type of material - master's thesis
Publication and manufacture - [Harbour Centre, Vancouver] : [J. Aleš], cop. 1992
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