Vertex-edge domination in cubic graphs

Abstract

We establish that any connected cubic graph of order n>6 has a minimum vertex-edge dominating set of at most 10n31 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 γve ratio is equal to 27. Finally, we show that the problem of determining the minimum γve-dominating set is NP-hard even in cubic planar graphs.