On Two Open Problems on Double Vertex-Edge Domination in Graphs

A vertex v of a graph G = ( V , E ) , ve-dominates every edge incident to v , as well as every edge adjacent to these incident edges. A set S ⊆ V is a double vertex-edge dominating set if every edge of E is ve-dominated by at least two vertices of S . The double vertex-edge domination number γ d v e ( G ) is the minimum cardinality of a double vertex-edge dominating set in G . A subset S ⊆ V is a total dominating set (respectively, a 2-dominating set) if every vertex in V has a neighbor in S (respectively, every vertex in V − S has at least two neighbors in S ). The total domination number γ t ( G ) is the minimum cardinality of a total dominating set of G , and the 2-domination number γ 2 ( G ) is the minimum cardinality of a 2-dominating set of G . Krishnakumari et al. (2017) showed that for every triangle-free graph G , γ d v e ( G ) ≤ γ 2 ( G ) , and in addition, if G has no isolated vertices, then γ d v e ( G ) ≤ γ t ( G ) . Moreover, they posed the problem of characterizing those graphs attaining the equality in the previous bounds. In this paper, we characterize all trees T with γ d v e ( T ) = γ t ( T ) or γ d v e ( T ) = γ 2 ( T ) .