Classes of Graphs for Which Upper Fractional Domination Equals Independence, Upper Domination, and upper Irredundance

This paper investigates cases where one graph parameter, upper fractional domination, is equal to three others: independence, upper domination and upper irredundance. We show that they are all equal for a large subclass, known as strongly perfect graphs, of the class of perfect graphs. They are also equal for odd cycles and upper bound graphs. However for simplicial graphs, upper irredundance might not equal the others, which are all equal. Also for many subclasses of perfect graphs other than the strongly perfect class, independence, upper domination and upper irredundance are not necessarily equal. We also show that if the graph join operation is used to combine two graphs which have some of the parameters equal, the resulting graph will have the same parameters equal.