Double vertex-edge domination

A vertex v of a graph G = (V,E) is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set (double vertex-edge dominating set, respectively) if every edge of E is ve-dominated by at least one vertex (at least two vertices) of S. The minimum cardinality of a vertex-edge dominating set (double vertex-edge dominating set, respectively) of G is the vertex-edge domination number γve(G) (the double vertex-edge domination number γdve(G), respectively). In this paper, we initiate the study of double vertex-edge domination. We first show that determining the number γdve(G) for bipartite graphs is NP-complete. We also prove that for every nontrivial connected graphs G, γdve(G) ≥ γve(G) + 1, and we characterize the trees T with γdve(T) = γve(T) + 1 or γdve(T) = γve(T) + 2. Finally, we provide two lower bounds on the double ve-domination number of trees and unicycle graphs in terms of the order n, the number of leaves and support vertices, and we characterize the trees attaining the lower bound.