Open irredundance and maximum degree in graphs

A necessary and sufficient condition for an open irredundant set of vertices of a graph to be maximal is obtained. This result is used to show that the smallest cardinality amongst the maximal open irredundant sets in an n-vertex isolate-free graph with maximum degree Δ is at least n(3Δ−1)/(2Δ3−5Δ2+8Δ−1) for Δ≥5, n/8 for Δ=4 and 2n/11 for Δ=3. The bounds are the best possible.