A note on independent vertex-edge domination in graphs

The independent vertex-edge domination number and the upper non-enclaving number of a graph G are denoted by ive(G) and Ψ(G), respectively. Boutrig et al. posed the following question: Let G be a connected graph with order n. Is Ψ(G)+ive(G)≤n?In this paper, we provide an infinite family of counterexamples. A new relationship between Ψ(G) and ive(G) is established. Furthermore, if G is a connected cubic graph, we answer this question in the affirmative.