Dynamical systems such as gene regulatory networks in which excitatory and inhibitory interactions occur among multiple elements are often studied with Boolean network models. In these models, dynamical states of individual components are described by binary values and interactions between components are represented by Boolean logic functions. The model system of interest here is a Boolean network model with an autonomous update scheme, in which different time delays are associated with different links of the network. In this modeling framework, adding one or two nodes and links to a network consisting of a ring of nodes can yield simple motifs that sustain stable oscillations. A recently published transcriptional oscillator underlying the yeast cell-cycle has a fixed point attractor for all proposed logic configurations, in contrast to most models of biological oscillators. It is a reasonable conjecture that the aforementioned simple motifs are the mechanism supporting the stable oscillations observed in the Boolean network representation of this yeast cell-cycle oscillator. Numerical analysis yields evidence supporting this hypothesis. This places the yeast cell-cycle oscillator in a different class than most biological oscillators in the literature, which relies on a frustrated logic.