Community structure is controlled, among multiple factors, by competition and predation. Using the R* rule and graphical analysis, we analyse here the feasibility, stability and assembly rules of resource-based food webs with up to three trophic levels. In particular, we show that (1) the stability of a food web with two plants and two generalist herbivores does not require that plants' resource exploitation abilities trade-off with resistance to the two herbivores, and (2) food webs with two plants and either one generalist herbivore and a carnivore or two generalist herbivores and two generalist carnivores are not feasible because of cascade competition between top consumers. The relative strength of species interactions and the relative impacts of plants and herbivores on factors which control their growth also play a critical role. We discuss how community structure constrains assembly rules and yields cascades of extinctions in food webs.