On Two Problems about \((0,2)\)-Graphs and Interval-Regular Graphs

We give operations on graphs preserving the property of being a \((0,2)\)-graph. In particular, these operations allow the construction of non-vertex-transitive \((0,2)\)-graphs. We also construct a family of regular interval-regular graphs which are not interval monotone, thus disproving a weaker version of a conjecture proposed by H.M. Mulder.