Discrete Mathematics ( IF 0.770 ) Pub Date : 2020-08-01 , DOI: 10.1016/j.disc.2020.112075
We establish that any connected cubic graph of order $n>6$ has a minimum vertex-edge dominating set of at most $\frac{10n}{31}$ vertices, thus affirmatively answering the open question posed by Klostermeyer et al. in Discussiones Mathematicae Graph Theory, https://doi.org/10.7151/dmgt.2175. On the other hand, we present an infinite family of cubic graphs whose ${\gamma }_{ve}$ ratio is equal to $\frac{2}{7}$. Finally, we show that the problem of determining the minimum ${\gamma }_{ve}$-dominating set is NP-hard even in cubic planar graphs.